 10.10.1.1: Suppose you start at the origin, move along the axis a dis 3, 7, ...
 10.10.2.1: Name all the equal vectors in the parallelogram shown. a
 10.10.8.1: Find the length of the curve.rt 2 sin t, 5t, 2 cos t 10 t 10 10.8
 10.10.3.1: Which of the following expressions are meaningful? Which u v u w ar...
 10.10.9.1: Find the velocity, acceleration, and speed of a particle with the g...
 10.1: What is the difference between a vector and a scalar?
 10.10.5.1: If , , and are noncoplanar vectors, let (These vectors occur in the...
 10.10.6.1: (a) What does the equation represent as a curve 2 y 0 in ? (b) What...
 10.10.1.2: Sketch the points , , , and on a single set of coordinate axes.
 10.10.2.2: Write each combination of vectors as a single vector. (a) PQl QRl (...
 10.10.8.2: Find the length of the curve.rt t 0 t 2 , sin t t cos t, cos t t si...
 10.10.3.2: Find the dot product of two vectors if their lengths are 6 and and ...
 10.10.9.2: Find the velocity, acceleration, and speed of a particle with the g...
 10.2: How do you add two vectors geometrically? How do you add them algeb...
 10.10.5.2: The line through the point and parallel to the vector
 10.10.6.2: (a) Sketch the graph of as a curve in . (b) Sketch the graph of as ...
 10.10.7.3: Find the limit.lim tl0 cos t, sin t, t ln t
 10.10.1.3: Which of the points , , and is closest to the plane? Which point l...
 10.10.2.3: Copy the vectors in the figure and use them to draw the following v...
 10.10.8.3: Find the length of the curve. rt i t 0 t 1 2 j t 3 3.
 10.10.3.3: Find a.b. a 6 b 5 a the angle between and is 23
 10.10.4.3: Find the cross product and verify that it is orthogonal to both a a...
 10.10.9.3: Find the velocity, acceleration, and speed of a particle with the g...
 10.3: If a is a vector and c is a scalar, how is ca related to a geometri...
 10.10.5.3: The line through the point and parallel to the vector
 10.10.6.3: Describe and sketch the surface.y 2 4z 2 4 z
 10.10.7.4: Find the limit.im tl arctan t, e2t , ln t t lim tl
 10.10.1.4: What are the projections of the point (2, 3, 5) on the , , and p...
 10.10.2.4: Copy the vectors in the figure and use them to draw the following v...
 10.10.8.4: Find the length of the curve.rt 12t i 8t 0 t 1 32 j 3t 2 k
 10.10.3.4: Find a.b. a 2, 3 b 0.7, 1.2 a
 10.10.4.4: Find the cross product and verify that it is orthogonal to both a a...
 10.10.9.4: Find the velocity, acceleration, and speed of a particle with the g...
 10.4: How do you find the vector from one point to another?
 10.10.5.4: The line through the origin and parallel to the line ,
 10.10.6.4: Describe and sketch the surface.z 4 x 2 y
 10.10.7.5: Sketch the curve with the given vector equation. Indicate with an a...
 10.10.1.5: Describe and sketch the surface in represented by the equation .
 10.10.2.5: Find a vector with representation given by the directed line segmen...
 10.10.8.5: Use Simpsons Rule with to estimate the length of the arc of the twi...
 10.10.3.5: Find a.b. a 4, 1, b 6, 3, 8 1 4
 10.10.4.5: Find the cross product and verify that it is orthogonal to both a a...
 10.10.9.5: Find the velocity, acceleration, and speed of a particle with the g...
 10.5: How do you find the dot product of two vectors if you know their le...
 10.10.5.5: The line through the point (1, 0, 6) and perpendicular to the plane
 10.10.6.5: Describe and sketch the surface.x y yz 4 2 0 z
 10.10.7.6: Sketch the curve with the given vector equation. Indicate with an a...
 10.10.1.6: (a) What does the equation represent in ? What does it represent in...
 10.10.2.6: Find a vector with representation given by the directed line segmen...
 10.10.8.6: Graph the curve with parametric equations , . Find the total length...
 10.10.3.6: Find a.b. a s, 2s, 3s b t, t, 5t a
 10.10.4.6: Find the cross product and verify that it is orthogonal to both a a...
 10.10.9.6: Find the velocity, acceleration, and speed of a particle with the g...
 10.6: How do you find the dot product of two vectors if you know their le...
 10.10.5.6: The line through the points and
 10.10.6.6: Describe and sketch the surface.yz 4 2
 10.10.7.7: Sketch the curve with the given vector equation. Indicate with an a...
 10.10.1.7: Find the lengths of the sides of the triangle . Is it a right trian...
 10.10.2.7: Find a vector with representation given by the directed line segmen...
 10.10.8.7: Reparametrize the curve with respect to arc length measured from th...
 10.10.3.7: Find a.b. a i 2 j 3k b 5i 9k a s
 10.10.4.7: Find the cross product and verify that it is orthogonal to both a a...
 10.10.9.7: Find the velocity, acceleration, and speed of a particle with the g...
 10.7: Write expressions for the scalar and vector projections of b onto a...
 10.10.5.7: The line through the points and
 10.10.6.7: Describe and sketch the surface.z cos x 2
 10.10.7.8: Sketch the curve with the given vector equation. Indicate with an a...
 10.10.1.8: Find the distance from to each of the following. (a) The plane (b)...
 10.10.2.8: Find a vector with representation given by the directed line segmen...
 10.10.8.8: Reparametrize the curve with respect to arc length measured from th...
 10.10.3.8: Find a.b. a 4 j 3k b 2i 4 j 6k a i
 10.10.4.8: If a i 2k and b j k, find a b. Sketch a, b, and a b as vectors star...
 10.10.9.8: Find the velocity, acceleration, and speed of a particle with the g...
 10.8: How do you find the cross product a b of two vectors if you know th...
 10.10.5.8: The line through and perpendicular to both and
 10.10.6.8: Describe and sketch the surface.x 2 y z cos x 2 1 x
 10.10.7.9: Sketch the curve with the given vector equation. Indicate with an a...
 10.10.1.9: Determine whether the points lie on straight line. A2, 4, 2 B3, 7, ...
 10.10.2.9: Find the sum of the given vectors and illustrate geometrically.3, 1...
 10.10.8.9: Suppose you start at the point and move 5 units along the curve , ,...
 10.10.3.9: If u is a unit vector, find and .
 10.10.4.9: State whether each expression is meaningful. If not, explain why. I...
 10.10.9.9: Find the velocity, acceleration, and speed of a particle with the g...
 10.9: How are cross products useful?
 10.10.5.9: The line through and parallel to the line
 10.10.6.9: (a) Find and identify the traces of the quadric surface and explain...
 10.10.7.10: Sketch the curve with the given vector equation. Indicate with an a...
 10.10.1.10: Find an equation of the sphere with center and radius 5. Describe i...
 10.10.2.10: Find the sum of the given vectors and illustrate geometrically.2, 1...
 10.10.8.10: Reparametrize the curve with respect to arc length measured from th...
 10.10.3.10: If u is a unit vector, find and .
 10.10.4.10: Find and determine whether u v is directed into the page or out of ...
 10.10.9.10: Find the velocity, acceleration, and speed of a particle with the g...
 10.10: (a) How do you find the area of the parallelogram determined by a a...
 10.10.5.10: The line of intersection of the planes and
 10.10.6.10: (a) Find and identify the traces of the quadric surface and explain...
 10.10.7.11: Sketch the curve with the given vector equation. Indicate with an a...
 10.10.1.11: Find an equation of the sphere that passes through the point and ha...
 10.10.2.11: Find the sum of the given vectors and illustrate geometrically.0, 1...
 10.10.8.11: (a) Find the unit tangent and unit normal vectors and . (b) Use For...
 10.10.3.11: (a) Show that i j j k k i 0 w u . (b) Show that i i j j k k 1 i j .
 10.10.4.11: Find and determine whether u v is directed into the page or out of ...
 10.10.9.11: Find the velocity and position vectors of a particle that has the g...
 10.11: How do you find a vector perpendicular to a plane?
 10.10.5.11: Is the line through and parallel to the line through and ?
 10.10.6.11: Find the traces of the given surface in the planes , , . Then ident...
 10.10.7.12: Sketch the curve with the given vector equation. Indicate with an a...
 10.10.1.12: Find an equation of the sphere that passes through the origin and w...
 10.10.2.12: Find the sum of the given vectors and illustrate geometrically.1, 0...
 10.10.8.12: (a) Find the unit tangent and unit normal vectors and . (b) Use For...
 10.10.3.12: A street vendor sells hamburgers, hot dogs, and soft drinks on a gi...
 10.10.4.12: The figure shows a vector in the plane and a vector in the directi...
 10.10.9.12: Find the velocity and position vectors of a particle that has the g...
 10.12: How do you find the angle between two intersecting planes?
 10.10.5.12: Is the line through and perpendicular to the line through and ?
 10.10.6.12: Find the traces of the given surface in the planes , , . Then ident...
 10.10.7.13: Find a vector equation and parametric equations for the line segmen...
 10.10.1.13: Show that the equation represents a sphere, and find its center and...
 10.10.2.13: Find a b, 2a 3b, , and . 1a 5, 12 b 3, 6 a
 10.10.8.13: (a) Find the unit tangent and unit normal vectors and . (b) Use For...
 10.10.4.13: If and , find and .a 1, 2, 1 b 0, 1, 3 a b
 10.10.9.13: (a) Find the position vector of a particle that has the given accel...
 10.13: Write a vector equation, parametric equations, and symmetric equati...
 10.10.5.13: (a) Find symmetric equations for the line that passes through the p...
 10.10.6.13: Find the traces of the given surface in the planes , , . Then ident...
 10.10.7.14: Find a vector equation and parametric equations for the line segmen...
 10.10.1.14: Show that the equation represents a sphere, and find its center and...
 10.10.2.14: Find a b, 2a 3b, , and . 1a 4 i j b i 2 j a
 10.10.8.14: (a) Find the unit tangent and unit normal vectors and . (b) Use For...
 10.10.4.14: If , , and , show that .
 10.10.9.14: (a) Find the position vector of a particle that has the given accel...
 10.14: Write a vector equation and a scalar equation for a plane.
 10.10.5.14: (a) Find parametric equations for the line through that is perpendi...
 10.10.6.14: Find the traces of the given surface in the planes , , . Then ident...
 10.10.7.15: Find a vector equation and parametric equations for the line segmen...
 10.10.1.15: Show that the equation represents a sphere, and find its center and...
 10.10.2.15: Find a b, 2a 3b, , and . 1a i 2 j 3k b 2 i j 5k a 4 i
 10.10.8.15: Use Theorem 10 to find the curvature.rt t 2 i t k rt
 10.10.4.15: Find two unit vectors orthogonal to both and .
 10.10.9.15: The position function of a particle is given by . When is the speed...
 10.15: (a) How do you tell if two vectors are parallel? (b) How do you tel...
 10.10.5.15: Find a vector equation for the line segment from to .
 10.10.6.15: Find the traces of the given surface in the planes , , . Then ident...
 10.10.7.16: Find a vector equation and parametric equations for the line segmen...
 10.10.1.16: Show that the equation represents a sphere, and find its center and...
 10.10.2.16: Find a b, 2a 3b, , and . 1a 2 i 4 j 4 k b 2 j k a i
 10.10.8.16: Use Theorem 10 to find the curvature.rt t i t j 1 t 2 k rt t
 10.10.3.16: Find, correct to the nearest degree, the three angles of the triang...
 10.10.4.16: Find two unit vectors orthogonal to both and .
 10.10.9.16: What force is required so that a particle of mass has the position ...
 10.16: (a) Describe a method for determining whether three points , , and ...
 10.10.5.16: Find parametric equations for the line segment from to .
 10.10.6.16: Find the traces of the given surface in the planes , , . Then ident...
 10.10.7.17: Match the parametric equations with the graphs (labeled IVI). Give ...
 10.10.1.17: (a) Prove that the midpoint of the line segment from to is x1 x2 2 ...
 10.10.2.17: Find a unit vector with the same direction as .
 10.10.8.17: Use Theorem 10 to find the curvature.rt 3t i 4 sin t j 4 cos t k rt
 10.10.3.17: Determine whether the given vectors are orthogonal, parallel, or ne...
 10.10.4.17: Show that for any vector in .
 10.10.9.17: A force with magnitude 20 N acts directly upward from the plane on...
 10.17: (a) How do you find the distance from a point to a line? (b) How do...
 10.10.5.17: Determine whether the lines and are parallel, skew, or intersecting...
 10.10.6.17: Find the traces of the given surface in the planes , , . Then ident...
 10.10.7.18: Match the parametric equations with the graphs (labeled IVI). Give ...
 10.10.1.18: Find an equation of a sphere if one of its diameters has endpoints ...
 10.10.2.18: Find a vector that has the same direction as but has length 6.
 10.10.8.18: Find the curvature of at the point (1, 0, 0).
 10.10.3.18: Determine whether the given vectors are orthogonal, parallel, or ne...
 10.10.4.18: Show that for all vectors and in .
 10.10.9.18: Show that if a particle moves with constant speed, then the velocit...
 10.18: What are the traces of a surface? How do you find them?
 10.10.5.18: Determine whether the lines and are parallel, skew, or intersecting...
 10.10.6.18: Find the traces of the given surface in the planes , , . Then ident...
 10.10.7.19: Match the parametric equations with the graphs (labeled IVI). Give ...
 10.10.1.19: Find equations of the spheres with center that touch (a) the plane...
 10.10.2.19: If lies in the first quadrant and makes an angle with the positive ...
 10.10.8.19: Find the curvature of at the point (1, 1, 1).
 10.10.3.19: Use vectors to decide whether the triangle with vertices , , and is...
 10.10.4.19: Prove Property 1 of Theorem 8.
 10.10.9.19: A projectile is fired with an initial speed of 500 ms and angle of ...
 10.19: Write equations in standard form of the six types of quadric surfaces.
 10.10.5.19: Determine whether the lines and are parallel, skew, or intersecting...
 10.10.6.19: Find the traces of the given surface in the planes , , . Then ident...
 10.10.7.20: Match the parametric equations with the graphs (labeled IVI). Give ...
 10.10.1.20: Find an equation of the largest sphere with center (5, 4, 9) that i...
 10.10.2.20: If a child pulls a sled through the snow with a force of 50 N exert...
 10.10.8.20: Graph the curve with parametric equations and find the curvature at...
 10.10.3.20: For what values of are the vectors and orthogonal?
 10.10.4.20: Prove Property 2 of Theorem 8.
 10.10.9.20: Rework Exercise 19 if the projectile is fired from a position 200 m...
 10.20: What is a vector function? How do you find its derivative and its i...
 10.10.5.20: Determine whether the lines and are parallel, skew, or intersecting...
 10.10.6.20: Find the traces of the given surface in the planes , , . Then ident...
 10.10.7.21: Match the parametric equations with the graphs (labeled IVI). Give ...
 10.10.1.21: Describe in words the region of represented by the equation or ineq...
 10.10.2.21: Two forces and with magnitudes 10 lb and 12 lb act on an object at ...
 10.10.8.21: Use Formula 11 to find the curvature.y xe y cos x x 1
 10.10.3.21: Find a unit vector that is orthogonal to both and .
 10.10.4.21: Prove Property 3 of Theorem 8.
 10.10.9.21: A ball is thrown at an angle of to the ground. If the ball lands 90...
 10.21: What is the connection between vector functions and space curves?
 10.10.5.21: The plane through the point and perpendicular to the vector 2, 1, 56
 10.10.6.21: Reduce the equation to one of the standard forms, classify the surf...
 10.10.7.22: Match the parametric equations with the graphs (labeled IVI). Give ...
 10.10.1.22: Describe in words the region of represented by the equation or ineq...
 10.10.2.22: Velocities have both direction and magnitude and thus are vectors. ...
 10.10.8.22: Use Formula 11 to find the curvature.y cos x x
 10.10.3.22: Find two unit vectors that make an angle of with .
 10.10.4.22: Prove Property 4 of Theorem 8.
 10.10.9.22: A gun is fired with angle of elevation . What is the muzzle speed i...
 10.22: (a) What is a smooth curve? (b) How do you find the tangent vector ...
 10.10.5.22: The plane through the point and with normal vector j 2k 4
 10.10.6.22: Reduce the equation to one of the standard forms, classify the surf...
 10.10.7.23: Show that the curve with parametric equations , , lies on the cone ...
 10.10.1.23: Describe in words the region of represented by the equation or ineq...
 10.10.2.23: A woman walks due west on the deck of a ship at 3 mih. The ship is ...
 10.10.8.23: Use Formula 11 to find the curvature.y 4x 52
 10.10.3.23: Find the scalar and vector projections of onto .a 3, 4 b 5, 0 b
 10.10.4.23: Find the area of the parallelogram with vertices , , , and .
 10.10.9.23: A gun has muzzle speed . Find two angles of elevation that can be u...
 10.23: If and are differentiable vector functions, is a scalar, and is a r...
 10.10.5.23: The plane through the origin and parallel to the plane 2x y 3z 1 j
 10.10.6.23: Reduce the equation to one of the standard forms, classify the surf...
 10.10.7.24: Show that the curve with parametric equations , , is the curve of i...
 10.10.1.24: Describe in words the region of represented by the equation or ineq...
 10.10.2.24: Ropes 3 m and 5 m in length are fastened to a holiday decoration th...
 10.10.8.24: At what point does the curve have maximum curvature? What happens t...
 10.10.3.24: Find the scalar and vector projections of onto .a 1, 2 b 4, 1 a
 10.10.4.24: Find the area of the parallelogram with vertices , , , and .
 10.10.9.24: A batter hits a baseball 3 ft above the ground toward the center fi...
 10.24: How do you find the length of a space curve given by a vector funct...
 10.10.5.24: The plane that contains the line , , and is parallel to the plane z...
 10.10.6.24: Reduce the equation to one of the standard forms, classify the surf...
 10.10.7.25: At what points does the curve intersect the paraboloid ?
 10.10.1.25: Describe in words the region of represented by the equation or ineq...
 10.10.2.25: A clothesline is tied between two poles, 8 m apart. The line is qui...
 10.10.8.25: At what point does the curve have maximum curvature? What happens t...
 10.10.3.25: Find the scalar and vector projections of onto .a 3, 6, 2 b 1, 2, 3 a
 10.10.4.25: (a) Find a vector orthogonal to the plane through the points , , an...
 10.10.9.25: A medieval city has the shape of a square and is protected by walls...
 10.25: (a) What is the definition of curvature? (b) Write a formula for cu...
 10.10.5.25: The plane through the points 25. 0, 1, 1 1, 0, 1 1, 1, 0 z 8
 10.10.6.25: Reduce the equation to one of the standard forms, classify the surf...
 10.10.6.26: Reduce the equation to one of the standard forms, classify the surf...
 10.10.7.26: Graph the curve with parametric equations Explain the appearance of...
 10.10.1.26: Describe in words the region of represented by the equation or ineq...
 10.10.2.26: The tension T at each end of the chain has magnitude 25 N. What is ...
 10.10.8.26: Find an equation of a parabola that has curvature 4 at the origin.
 10.10.3.26: Find the scalar and vector projections of onto .a i j k b i j k a 3,
 10.10.4.26: (a) Find a vector orthogonal to the plane through the points , , an...
 10.10.9.26: A ball with mass 0.8 kg is thrown southward into the air with a spe...
 10.26: Write formulas for the unit normal and binormal vectors of a smooth...
 10.10.5.26: The plane through the origin and the points and 5, 1, 3 2,
 10.10.6.27: Reduce the equation to one of the standard forms, classify the surf...
 10.10.7.27: Show that the curve with parametric equations , , passes through th...
 10.10.1.27: Describe in words the region of represented by the equation or ineq...
 10.10.2.27: (a) Draw the vectors , , and (b) Show, by means of a sketch, that t...
 10.10.8.27: (a) Is the curvature of the curve shown in the figure greater at or...
 10.10.3.27: Show that the vector is orthogonal to . (It is called an orthogonal...
 10.10.4.27: (a) Find a vector orthogonal to the plane through the points , , an...
 10.10.9.27: Water traveling along a straight portion of a river normally flows ...
 10.27: (a) How do you find the velocity, speed, and acceleration of a part...
 10.10.5.27: The plane that passes through the point and contains the line x 4 2...
 10.10.6.28: Reduce the equation to one of the standard forms, classify the surf...
 10.10.7.28: Find a vector function that represents the curve of intersection of...
 10.10.1.28: Describe in words the region of represented by the equation or ineq...
 10.10.2.28: Suppose that and are nonzero vectors that are not parallel and is a...
 10.10.8.28: Use a graphing calculator or computer to graph both the curve and i...
 10.10.3.28: For the vectors in Exercise 24, find and illustrate by drawing the ...
 10.10.4.28: (a) Find a vector orthogonal to the plane through the points , , an...
 10.10.9.28: Find the tangential and normal components of the acceleration vecto...
 10.28: State Keplers Laws.
 10.10.5.28: The plane that passes through the point and contains the line with ...
 10.10.6.29: Sketch the region bounded by the surfaces and for .
 10.10.7.29: Find a vector function that represents the curve of intersection of...
 10.10.1.29: Describe in words the region of represented by the equation or ineq...
 10.10.2.29: If and , describe the set of all points such that .
 10.10.8.29: Use a graphing calculator or computer to graph both the curve and i...
 10.10.3.29: If , find a vector such that .
 10.10.4.29: Find the volume of the parallelepiped determined by the vectors , ,...
 10.10.9.29: Find the tangential and normal components of the acceleration vecto...
 10.29: Identify and sketch the graph of each surface.x 2 y 2 4z 2 y
 10.10.5.29: The plane that passes through the point and contains the line of in...
 10.10.6.30: Sketch the region bounded by the paraboloids and .
 10.10.7.30: Find a vector function that represents the curve of intersection of...
 10.10.1.30: Describe in words the region of represented by the equation or ineq...
 10.10.2.30: If , , and , describe the set of all points such that , where .
 10.10.8.30: Two graphs, and , are shown. One is a curve and the other is the gr...
 10.10.3.30: Suppose that and are nonzero vectors. (a) Under what circumstances ...
 10.10.4.30: Find the volume of the parallelepiped determined by the vectors , ,...
 10.10.9.30: Find the tangential and normal components of the acceleration vecto...
 10.30: Identify and sketch the graph of each surface.4x y 2z 4 x
 10.10.5.30: The plane that passes through the line of intersection of the plane...
 10.10.6.31: Find an equation for the surface consisting of all points that are ...
 10.10.7.31: Try to sketch by hand the curve of intersection of the circular cyl...
 10.10.1.31: Write inequalities to describe the region.The halfspace consisting...
 10.10.2.31: Figure 16 gives a geometric demonstration of Property 2 of vectors....
 10.10.8.31: Two graphs, and , are shown. One is a curve and the other is the gr...
 10.10.3.31: A constant force with vector representation F 10 i 18 j 6k moves an...
 10.10.4.31: Find the volume of the parallelepiped with adjacent edges , , and ....
 10.10.9.31: Find the tangential and normal components of the acceleration vecto...
 10.31: Identify and sketch the graph of each surface.4x 2 y 2 4z 2 4 4x
 10.10.5.31: Find the point at which the line , , intersects the plane .
 10.10.6.32: Find an equation for the surface consisting of all points for which...
 10.10.7.32: Try to sketch by hand the curve of intersection of the parabolic cy...
 10.10.1.32: Write inequalities to describe the region.The solid rectangular box...
 10.10.2.32: Prove Property 5 of vectors algebraically for the case . Then use s...
 10.10.8.32: Use Theorem 10 to show that the curvature of a plane parametric cur...
 10.10.3.32: Find the work done by a force of 20 lb acting in the direction N W ...
 10.10.4.32: Find the volume of the parallelepiped with adjacent edges , , and ....
 10.10.9.32: If a particle with mass moves with position vector , then its angul...
 10.32: Identify and sketch the graph of each surface.y 2 z 2 1 x 2 4x
 10.10.5.32: Where does the line through and intersect the plane ?
 10.10.6.33: Graph the surfaces and on a common screen using the domain , and ob...
 10.10.1.33: Write inequalities to describe the region.The region consisting of ...
 10.10.2.33: Use vectors to prove that the line joining the midpoints of two sid...
 10.10.8.33: Use the formula in Exercise 32 to find the curvature. y et x e sin ...
 10.10.3.33: A woman exerts a horizontal force of 25 lb on a crate as she pushes...
 10.10.4.33: Use the scalar triple product to verify that the vectors , , and ar...
 10.10.9.33: The position function of a spaceship is and the coordinates of a sp...
 10.33: Identify and sketch the graph of each surface.4x 2 4y 2 8y z 2 0 y 2
 10.10.5.33: Determine whether the planes are parallel, perpendicular, or neithe...
 10.10.6.34: Show that the curve of intersection of the surfaces and lies in a p...
 10.10.1.34: Write inequalities to describe the region.The solid upper hemispher...
 10.10.8.34: Use the formula in Exercise 32 to find the curvature. 2 x 1 t 3 y t...
 10.10.3.34: A wagon is pulled a distance of 100 m along a horizontal path by a ...
 10.10.4.34: Use the scalar triple product to determine whether the points , , ,...
 10.10.9.34: A rocket burning its onboard fuel while moving through space has ve...
 10.34: Identify and sketch the graph of each surface.x y2 z2 2y 4z 5 4x 2
 10.10.5.34: Determine whether the planes are parallel, perpendicular, or neithe...
 10.10.1.35: Find an equation of the set of all points equidistant from the poin...
 10.10.8.35: Find the vectors , , and at the given point.rt t , 1) 2
 10.10.3.35: Use a scalar projection to show that the distance from a point to t...
 10.10.4.35: A bicycle pedal is pushed by a foot with a 60N force as shown. The...
 10.35: An ellipsoid is created by rotating the ellipse about the axis. Fi...
 10.10.5.35: Determine whether the planes are parallel, perpendicular, or neithe...
 10.10.1.36: Find the volume of the solid that lies inside both of the spheres a...
 10.10.8.36: Find the vectors , , and at the given point.rt e 1, 0, 1 t ,
 10.10.3.36: If , and , show that the vector equation represents a sphere, and f...
 10.10.4.36: Find the magnitude of the torque about if a 36lb force is applied ...
 10.36: A surface consists of all points such that the distance from to the...
 10.10.5.36: Determine whether the planes are parallel, perpendicular, or neithe...
 10.10.8.37: Find equations of the normal plane and osculating plane of the curv...
 10.10.3.37: Find the angle between a diagonal of a cube and one of its edges.
 10.10.4.37: A wrench 30 cm long lies along the positive axis and grips a bolt ...
 10.37: (a) Sketch the curve with vector function (b) Find and .
 10.10.5.37: (a) Find symmetric equations for the line of intersection of the pl...
 10.10.8.38: Find equations of the normal plane and osculating plane of the curv...
 10.10.3.38: Find the angle between a diagonal of a cube and a diagonal of one o...
 10.10.4.38: Let v 5j and let u be a vector with length 3 that starts at the ori...
 10.38: Let . (a) Find the domain of . (b) Find . (c) Find .
 10.10.5.38: Find an equation for the plane consisting of all points that are eq...
 10.10.7.39: Find the derivative of the vector function.rt t rt cos 3t, t, sin 3...
 10.10.8.39: Find equations of the osculating circles of the ellipse at the poin...
 10.10.3.39: A molecule of methane, , is structured with the four hydrogen atoms...
 10.10.4.39: (a) Let be a point not on the line that passes through the points a...
 10.39: Find a vector function that represents the curve of intersection of...
 10.10.5.39: Find an equation of the plane with intercept , intercept , and i...
 10.10.7.40: Find the derivative of the vector function.rt cos 3t, t, sin 3t 2
 10.10.8.40: Find equations of the osculating circles of the parabola at the poi...
 10.10.3.40: If , where , , and are all nonzero vectors, show that bisects the a...
 10.10.4.40: (a) Let be a point not on the plane that passes through the points ...
 10.40: Find parametric equations for the tangent line to the curve , , at ...
 10.10.5.40: (a) Find the point at which the given lines intersect: (b) Find an ...
 10.10.7.41: Find the derivative of the vector function.rt et 2 i j ln1 3t k rt t
 10.10.8.41: At what point on the curve , , is the normal plane parallel to the ...
 10.10.3.41: Prove Properties 2, 4, and 5 of the dot product (Theorem 2)
 10.10.4.41: Prove that .
 10.10.5.41: Find parametric equations for the line through the point that is pa...
 10.10.7.42: Find the derivative of the vector function.rt at cos 3t i b sin3 t ...
 10.10.8.42: Is there a point on the curve in Exercise 41 where the osculating p...
 10.10.3.42: Suppose that all sides of a quadrilateral are equal in length and o...
 10.10.4.42: Prove Property 6 of Theorem 8, that is,a b c a cb a bc 41. a b
 10.10.5.42: Find parametric equations for the line through the point that is pe...
 10.10.7.43: Find the derivative of the vector function.rt a t b t 2 43. c rt
 10.10.8.43: Show that the curvature is related to the tangent and normal vector...
 10.10.3.43: Use Theorem 3 to prove the CauchySchwarz Inequality:a b a b
 10.10.4.43: Use Exercise 42 to prove thata b c b c a c a b 0 a b
 10.10.5.43: Which of the following four planes are parallel? Are any of them id...
 10.10.7.44: Find the derivative of the vector function.rt t a b t c rt a
 10.10.8.44: Show that the curvature of a plane curve is , where is the angle be...
 10.10.3.44: The Triangle Inequality for vectors is (a) Give a geometric interpr...
 10.10.4.44: Prove thata b c d a c a d b c b d a b c
 10.10.5.44: Which of the following four lines are parallel? Are any of them ide...
 10.10.7.45: Find the unit tangent vector at the point with the given value of t...
 10.10.8.45: (a) Show that is perpendicular to . (b) Show that is perpendicular ...
 10.10.3.45: The Parallelogram Law states that (a) Give a geometric interpretati...
 10.10.4.45: Suppose that . (a) If , does it follow that ? (b) If , does it foll...
 10.10.5.45: Use the formula in Exercise 39 in Section 10.4 to find the distance...
 10.10.7.46: Find the unit tangent vector at the point with the given value of t...
 10.10.8.46: The following formulas, called the FrenetSerret formulas, are of f...
 10.10.4.46: If , , and are noncoplanar vectors, let (These vectors occur in the...
 10.10.5.46: Use the formula in Exercise 39 in Section 10.4 to find the distance...
 10.10.7.47: Find parametric equations for the tangent line to the curve with th...
 10.10.8.47: Use the FrenetSerret formulas to prove each of the following. (Pri...
 10.10.5.47: Find the distance from the point to the given plane. 2, 8, 5 x 2y 2...
 10.10.7.48: Find parametric equations for the tangent line to the curve with th...
 10.10.8.48: Show that the circular helix x where and are positive constants, ha...
 10.10.5.48: Find the distance from the point to the given plane. 3, 2, 7 4x 6y ...
 10.10.7.49: Find parametric equations for the tangent line to the curve with th...
 10.10.8.49: The DNA molecule has the shape of a double helix (see Figure 3 on p...
 10.10.5.49: Find the distance between the given parallel planes.z x 2y 1 3x 6y ...
 10.10.7.50: Find parametric equations for the tangent line to the curve with th...
 10.10.8.50: Lets consider the problem of designing a railroad track to make a s...
 10.10.5.50: Find the distance between the given parallel planes.3x 6y 9z 4 x 2y...
 10.10.7.51: Find parametric equations for the tangent line to the curve with th...
 10.10.5.51: Show that the distance between the parallel planes and is
 10.10.7.52: Find parametric equations for the tangent line to the curve with th...
 10.10.5.52: Find equations of the planes that are parallel to the plane and two...
 10.10.7.53: Determine whether the curve is smooth. (a) (b) (c)
 10.10.5.53: Show that the lines with symmetric equations and are skew, and find...
 10.10.7.54: a) Find the point of intersection of the tangent lines to the curve...
 10.10.5.54: Find the distance between the skew lines with parametric equations ...
 10.10.7.55: The curves and intersect at the origin. Find their angle of interse...
 10.10.5.55: If , , and are not all 0, show that the equation represents a plane...
 10.10.7.56: At what point do the curves and intersect? Find their angle of inte...
 10.10.5.56: Give a geometric description of each family of planes. (a) (b) (c) ...
 10.10.7.57: Evaluate the integral.y 1 0 16t 3 i 9t 2 j 25t 4 k dt r2s
 10.10.7.58: Evaluate the integral.y 1 0 4 1 t 2 j 2t 1 t 2 k dt y
 10.10.7.59: Evaluate the integral.y 2 0 3 sin2 t cos t i 3 sin t cos 2 t j 2 si...
 10.10.7.60: Evaluate the integral. y 2 1 (t 2 i tst 1 j t sin t k) dt y
 10.10.7.61: Evaluate the integral.et i 2t j ln t k dt y
 10.10.7.62: Evaluate the integral.y cos t i sin t j t k dt y e
 10.10.7.63: Find if and rt 2t i 3t r1 i j 2 rt j st k y cos t i
 10.10.7.64: Find if and rt t i e r0 i j k t j te t rt k rt 2t i 3
 10.10.7.65: If two objects travel through space along two different curves, its...
 10.10.7.66: Two particles travel along the space curves Do the particles collid...
 10.10.7.67: Suppose and are vector functions that possess limits as and let be ...
 10.10.7.68: Show that if and only if for every there is a number such that when...
 10.10.7.69: Prove Formula 1 of Theorem 5.
 10.10.7.70: Prove Formula 3 of Theorem 5.
 10.10.7.71: Prove Formula 5 of Theorem 5.
 10.10.7.72: Prove Formula 6 of Theorem 5.
 10.10.7.73: If and find .
 10.10.7.74: If and are the vector functions in Exercise 73, find .
 10.10.7.75: Show that if is a vector function such that exists, then d dt rt rt...
 10.10.7.76: Find an expression for .
 10.10.7.77: If , show that . [Hint: ]
 10.10.7.78: If a curve has the property that the position vector is always perp...
 10.10.7.79: If , show that ut rt rt rt ut rt
Solutions for Chapter 10: VECTORS AND THE GEOMETRY OF SPACE
Full solutions for Essential Calculus (Available Titles CengageNOW)  1st Edition
ISBN: 9780495014423
Solutions for Chapter 10: VECTORS AND THE GEOMETRY OF SPACE
Get Full SolutionsThis textbook survival guide was created for the textbook: Essential Calculus (Available Titles CengageNOW), edition: 1. Chapter 10: VECTORS AND THE GEOMETRY OF SPACE includes 440 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 440 problems in chapter 10: VECTORS AND THE GEOMETRY OF SPACE have been answered, more than 16587 students have viewed full stepbystep solutions from this chapter. Essential Calculus (Available Titles CengageNOW) was written by and is associated to the ISBN: 9780495014423.

Absolute value of a vector
See Magnitude of a vector.

Blind experiment
An experiment in which subjects do not know if they have been given an active treatment or a placebo

Division
a b = aa 1 b b, b Z 0

Elimination method
A method of solving a system of linear equations

Exponential function
A function of the form ƒ(x) = a ? bx,where ?0, b > 0 b ?1

Exponential growth function
Growth modeled by ƒ(x) = a ? b a > 0, b > 1 .

Extracting square roots
A method for solving equations in the form x 2 = k.

Factor
In algebra, a quantity being multiplied in a product. In statistics, a potential explanatory variable under study in an experiment, .

Linear correlation
A scatter plot with points clustered along a line. Correlation is positive if the slope is positive and negative if the slope is negative

Mapping
A function viewed as a mapping of the elements of the domain onto the elements of the range

Nappe
See Right circular cone.

Nonsingular matrix
A square matrix with nonzero determinant

Paraboloid of revolution
A surface generated by rotating a parabola about its line of symmetry.

Parallel lines
Two lines that are both vertical or have equal slopes.

Polar coordinate system
A coordinate system whose ordered pair is based on the directed distance from a central point (the pole) and the angle measured from a ray from the pole (the polar axis)

Positive angle
Angle generated by a counterclockwise rotation.

Pythagorean
Theorem In a right triangle with sides a and b and hypotenuse c, c2 = a2 + b2

Root of an equation
A solution.

Standard form of a polar equation of a conic
r = ke 1 e cos ? or r = ke 1 e sin ? ,

Vertical line test
A test for determining whether a graph is a function.