 12.12.1: What surfaces in are represented by the following equations?
 12.12.2: Describe and sketch the surface in represented by the equation y=x
 12.12.3: The distance from the point to the point is PQ s1 2s1 4 4 3 2 3 12 ...
 12.12.4: Find an equation of a sphere with radius and center Ch, k, l
 12.12.5: Show that is the equation of a sphere, and find its center and radius.
 12.12.6: What region in is represented by the following inequalities?1 x z 0...
 12.12.7: Suppose you start at the origin, move along the axis a distance of...
 12.12.8: Sketch the points , , , and on a single set of coordinate axes
 12.12.9: Which of the points , , and is closest to the plane? Which point l...
 12.12.10: What are the projections of the point (2, 3, 5) on the , , and p...
 12.12.11: Describe and sketch the surface in represented by the equation x y 2
 12.12.12: a) What does the equation represent in ? What does it represent in ...
 12.12.13: Find the lengths of the sides of the triangle . Is it a right trian...
 12.12.14: Find the lengths of the sides of the triangle . Is it a right trian...
 12.12.15: Determine whether the points lie on straight line.
 12.12.16: Find the distance from to each of the following. (a) The plane (b)...
 12.12.17: Find an equation of the sphere with center and radius 5. What is th...
 12.12.18: Find an equation of the sphere with center and radius 5. Describe i...
 12.12.19: Find an equation of the sphere that passes through the point and ha...
 12.12.20: Find an equation of the sphere that passes through the origin and w...
 12.12.21: Show that the equation represents a sphere, and find its center and...
 12.12.22: Show that the equation represents a sphere, and find its center and...
 12.12.23: Show that the equation represents a sphere, and find its center and...
 12.12.24: Show that the equation represents a sphere, and find its center and...
 12.12.25: (a) Prove that the midpoint of the line segment from to is (b) Find...
 12.12.26: Find an equation of a sphere if one of its diameters has endpoints ...
 12.12.27: Find equations of the spheres with center that touch (a) the plane...
 12.12.28: Find an equation of the largest sphere with center (5, 4, 9) that i...
 12.12.29: Describe in words the region of represented by the equation or ineq...
 12.12.30: Describe in words the region of represented by the equation or ineq...
 12.12.31: Describe in words the region of represented by the equation or ineq...
 12.12.32: Describe in words the region of represented by the equation or ineq...
 12.12.33: Describe in words the region of represented by the equation or ineq...
 12.12.34: Describe in words the region of represented by the equation or ineq...
 12.12.35: Describe in words the region of represented by the equation or ineq...
 12.12.36: Describe in words the region of represented by the equation or ineq...
 12.12.37: Describe in words the region of represented by the equation or ineq...
 12.12.38: Describe in words the region of represented by the equation or ineq...
 12.12.39: Write inequalities to describe the region The region between the p...
 12.12.40: Write inequalities to describe the region The solid cylinder that l...
 12.12.41: Write inequalities to describe the region The region consisting of ...
 12.12.42: Write inequalities to describe the region The solid upper hemispher...
 12.12.43: The figure shows a line in space and a second line L2 which is the ...
 12.12.44: Consider the points such that the distance from to is twice the dis...
 12.12.45: Find an equation of the set of all points equidistant from the poin...
 12.12.46: Find the volume of the solid that lies inside both of the spheres x...
 12.12.47: Draw the sum of the vectors shown in Figure 5
 12.12.48: If are the vectors shown in Figure 9, draw a2b
 12.12.49: Find the vector represented by the directed line segment with initi...
 12.12.50: If and , find and the vectors a+b ab,3b and 2a+5b
 12.12.51: Find the unit vector in the direction of the vector 2i j 2k
 12.12.52: A 100lb weight hangs from two wires as shown in Figure 19. Find th...
 12.12.53: Are the following quantities vectors or scalars? Explain (a) The co...
 12.12.54: What is the relationship between the point (4, 7) and the vector ? ...
 12.12.55: Name all the equal vectors in the parallelogram shown.
 12.12.56: Write each combination of vectors as a single vector. (a) PQl QRl (...
 12.12.57: Copy the vectors in the figure and use them to draw the following v...
 12.12.58: Copy the vectors in the figure and use them to draw the following v...
 12.12.59: Find a vector with representation given by the directed line segmen...
 12.12.60: Find a vector with representation given by the directed line segmen...
 12.12.61: Find a vector with representation given by the directed line segmen...
 12.12.62: Find a vector with representation given by the directed line segmen...
 12.12.63: Find a vector with representation given by the directed line segmen...
 12.12.64: Find a vector with representation given by the directed line segmen...
 12.12.65: Find the sum of the given vectors and illustrate geometrically1, 46...
 12.12.66: Find the sum of the given vectors and illustrate geometrically2, 15...
 12.12.67: Find the sum of the given vectors and illustrate geometrically0, 1,...
 12.12.68: Find the sum of the given vectors and illustrate geometrically1, 0,...
 12.12.69: Find a b, 2a 3b, , and .a b a 5, 12b 3, 6 a a b
 12.12.70: Find a b, 2a 3b, , and .a b a 4 i j b i 2 j a
 12.12.71: Find a b, 2a 3b, , and .a b a i 2 j 3k b 2 i j 5k a 4 i
 12.12.72: Find a b, 2a 3b, , and .a b a 2 i 4 j 4 k b 2 j k a i
 12.12.73: Find a unit vector that has the same direction as the given vector3...
 12.12.74: Find a unit vector that has the same direction as the given vector4...
 12.12.75: Find a unit vector that has the same direction as the given vector8...
 12.12.76: Find a vector that has the same direction as but has length 6.
 12.12.77: If lies in the first quadrant and makes an angle with the positive ...
 12.12.78: If a child pulls a sled through the snow on a level path with a for...
 12.12.79: A quarterback throws a football with angle of elevation and speed ....
 12.12.80: Find the magnitude of the resultant force and the angle it makes wi...
 12.12.81: Find the magnitude of the resultant force and the angle it makes wi...
 12.12.82: The magnitude of a velocity vector is called speed. Suppose that a ...
 12.12.83: A woman walks due west on the deck of a ship at 3 mih. The ship is ...
 12.12.84: Ropes 3 m and 5 m in length are fastened to a holiday decoration th...
 12.12.85: A clothesline is tied between two poles, 8 m apart. The line is qui...
 12.12.86: The tension T at each end of the chain has magnitude 25 N. What is ...
 12.12.87: Find the unit vectors that are parallel to the tangent line to the ...
 12.12.88: (a) Find the unit vectors that are parallel to the tangent line to ...
 12.12.89: If , , and are the vertices of a triangle, find ABl BCl CAl.
 12.12.90: Let be the point on the line segment that is twice as far from as i...
 12.12.91: (a) Draw the vectors , , and (b) Show, by means of a sketch, that t...
 12.12.92: Suppose that and are nonzero vectors that are not parallel and is a...
 12.12.93: If and , describe the set of all points such that r r0 1 4
 12.12.94: If , , and , describe the set of all points such that , where k r1 r2
 12.12.95: . Figure 16 gives a geometric demonstration of Property 2 of vector...
 12.12.96: Prove Property 5 of vectors algebraically for the case . Then use s...
 12.12.97: Use vectors to prove that the line joining the midpoints of two sid...
 12.12.98: Suppose the three coordinate planes are all mirrored and a light ra...
 12.12.99: If the vectors a and b have lengths 4 and 6, and the angle between ...
 12.12.100: Find the angle between the vectors and b 5, 3, 2 cos
 12.12.101: Show that is perpendicular to 5i 4j 2k
 12.12.102: Find the direction angles of the vector a 1, 2, 3a
 12.12.103: A wagon is pulled a distance of 100 m along a horizontal path by a ...
 12.12.104: A force is given by a vector and moves a particle from the point to...
 12.12.105: Which of the following expressions are meaningful? Which are u v u ...
 12.12.106: Find the dot product of two vectors if their lengths are 6 and and ...
 12.12.107: Find a.b.
 12.12.108: Find a.b.a 2, 3b 0.7, 1.2 a 2
 12.12.109: Find a.b.a 2, 3b 0.7, 1.2 a
 12.12.110: Find a.b.a 4, 1, b 6, 3, 81 4
 12.12.111: Find a.b.a s, 2s, 3sb t, t, 5t a 4,
 12.12.112: Find a.b.a i 2 j 3k b 5i 9k a
 12.12.113: Find a.b.a 4 j 3k b 2i 4 j 6k a
 12.12.114: Find a.b.a 6 b 5 b
 12.12.115: Find a.b.a 3 b s6
 12.12.116: If u is a unit vector, find and u.w
 12.12.117: (a) Show thati.j=j.k=k.i=0 . (b) Show that i.i.=j.j=k.k=1.
 12.12.118: A street vendor sells hamburgers, hot dogs, and soft drinks on a gi...
 12.12.119: Find the angle between the vectors. (First find an exactexpression ...
 12.12.120: Find the angle between the vectors. (First find an exactexpression ...
 12.12.121: Find the angle between the vectors. (First find an exactexpression ...
 12.12.122: Find the angle between the vectors. (First find an exactexpression ...
 12.12.123: Find the angle between the vectors. (First find an exactexpression ...
 12.12.124: Find the angle between the vectors. (First find an exactexpression ...
 12.12.125: Find, correct to the nearest degree, the three angles of the triang...
 12.12.126: Find, correct to the nearest degree, the three angles of the triang...
 12.12.127: Determine whether the given vectors are orthogonal, parallel, or ne...
 12.12.128: Determine whether the given vectors are orthogonal, parallel, or ne...
 12.12.129: Use vectors to decide whether the triangle with vertices , , and is...
 12.12.130: For what values of are the vectors and orthogonal?
 12.12.131: Find a unit vector that is orthogonal to both i+j and i+k
 12.12.132: Find two unit vectors that make an angle of with . 2933 Find the di...
 12.12.133: Find two unit vectors that make an angle of with v=(3,4)
 12.12.134: Find the direction cosines and direction angles of the vector. (Giv...
 12.12.135: Find the direction cosines and direction angles of the vector. (Giv...
 12.12.136: Find the direction cosines and direction angles of the vector. (Giv...
 12.12.137: Find the direction cosines and direction angles of the vector. (Giv...
 12.12.138: Find the direction cosines and direction angles of the vector. (Giv...
 12.12.139: If a vector has direction angles and , find the third direction ang...
 12.12.140: Find the scalar and vector projections of b onto a a 3, 4b 5, 0 ab
 12.12.141: Find the scalar and vector projections of b onto aa 1, 2b 4, 1 a 3,
 12.12.142: Find the scalar and vector projections of b onto aa 3, 6, 2b 1, 2, ...
 12.12.143: Find the scalar and vector projections of b onto aa 2, 3, 6, b 5, 1...
 12.12.144: Find the scalar and vector projections of b onto a a 2i j 4k 2k
 12.12.145: Find the scalar and vector projections of b onto a a i j k b i j k b j
 12.12.146: Show that the vector is orthogonal to a
 12.12.147: For the vectors in Exercise 36, find and illustrate by drawing the ...
 12.12.148: f , find a vector such that compa b 2
 12.12.149: Suppose that and are nonzero vectors. (a) Under what circumstances ...
 12.12.150: Find the work done by a force that moves an object from the point t...
 12.12.151: A tow truck drags a stalled car along a road. The chain makes an an...
 12.12.152: A sled is pulled along a level path through snow by a rope. A 30lb...
 12.12.153: A boat sails south with the help of a wind blowing in the direction...
 12.12.154: Use a scalar projection to show that the distance from a point to t...
 12.12.155: If , and , show that the vector equation represents a sphere, and f...
 12.12.156: Find the angle between a diagonal of a cube and one of its edges.
 12.12.157: Find the angle between a diagonal of a cube and a diagonal of one o...
 12.12.158: A molecule of methane, , is structured with the four hydrogen atoms...
 12.12.159: If , where , , and are all nonzero vectors, show that bisects the a...
 12.12.160: Prove Properties 2, 4, and 5 of the dot product (Theorem 2)
 12.12.161: Suppose that all sides of a quadrilateral are equal in length and o...
 12.12.162: Use Theorem 3 to prove the CauchySchwarz Inequality:a b a b
 12.12.163: The Triangle Inequality for vectors is (a) Give a geometric interpr...
 12.12.164: The Parallelogram Law states that (a) Give a geometric interpretati...
 12.12.165: Show that if and are orthogonal, then the vectors and must have the...
 12.12.166: Show that if and are orthogonal, then the vectors and must have the...
 12.12.167: Find the cross product and verify that it is orthogonal to both a a...
 12.12.168: Find the cross product and verify that it is orthogonal to both a a...
 12.12.169: Find the cross product and verify that it is orthogonal to both a a...
 12.12.170: Find the cross product and verify that it is orthogonal to both a a...
 12.12.171: Find the cross product and verify that it is orthogonal to both a a...
 12.12.172: Find the cross product and verify that it is orthogonal to both a a...
 12.12.173: Find the cross product and verify that it is orthogonal to both a a...
 12.12.174: If a i 2k and b j k, find a b. Sketch a, b, and a b as vectors star...
 12.12.175: Find the vector, not with determinants, but by using properties of ...
 12.12.176: Find the vector, not with determinants, but by using properties of ...
 12.12.177: Find the vector, not with determinants, but by using properties of ...
 12.12.178: Find the vector, not with determinants, but by using properties of ...
 12.12.179: State whether each expression is meaningful. If not, explain why. I...
 12.12.180: State whether each expression is meaningful. If not, explain why. I...
 12.12.181: State whether each expression is meaningful. If not, explain why. I...
 12.12.182: State whether each expression is meaningful. If not, explain why. I...
 12.12.183: State whether each expression is meaningful. If not, explain why. I...
 12.12.184: State whether each expression is meaningful. If not, explain why. I...
 12.12.185: Find and determine whether u v is directed into the page or out of ...
 12.12.186: Find and determine whether u v is directed into the page or out of ...
 12.12.187: The figure shows a vector in the plane and a vector in the directi...
 12.12.188: If a 1, 2, 1b 0 and b 0, 1, 3a , find b 0, 1, 3a and b a
 12.12.189: If , a 3, 1, 2b 1, 1, 0c 0, 0, and c 0, 0, 4, show that a b ca bc
 12.12.190: Find two unit vectors orthogonal to both and 0, 4, 4.
 12.12.191: Find two unit vectors orthogonal to both P and
 12.12.192: Show that for any vector in V2
 12.12.193: Show that for all vectors and in V3
 12.12.194: Prove Property 1 of Theorem 8
 12.12.195: Prove Property 2 of Theorem 8
 12.12.196: Prove Property 3 of Theorem 8.
 12.12.197: Prove Property 4 of Theorem 8
 12.12.198: Find the area of the parallelogram with vertices , , , and
 12.12.199: Find the area of the parallelogram with vertices , , , and
 12.12.200: a) Find a nonzero vector orthogonal to the plane through the points...
 12.12.201: a) Find a nonzero vector orthogonal to the plane through the points...
 12.12.202: a) Find a nonzero vector orthogonal to the plane through the points...
 12.12.203: a) Find a nonzero vector orthogonal to the plane through the points...
 12.12.204: Find the volume of the parallelepiped determined by the vectors , ,...
 12.12.205: Find the volume of the parallelepiped determined by the vectors , ,...
 12.12.206: Find the volume of the parallelepiped with adjacent edges , , and P...
 12.12.207: Find the volume of the parallelepiped with adjacent edges , , and P...
 12.12.208: Use the scalar triple product to verify that the vectors , , and ar...
 12.12.209: Use the scalar triple product to determine whether the points , , ,...
 12.12.210: A bicycle pedal is pushed by a foot with a 60N force as shown. The...
 12.12.211: Find the magnitude of the torque about if a 36lb force is applied ...
 12.12.212: A wrench 30 cm long lies along the positive axis and grips a bolt ...
 12.12.213: Let v 5j and let u be a vector with length 3 that starts at the ori...
 12.12.214: a) Let be a point not on the line that passes through the points an...
 12.12.215: (a) Let be a point not on the plane that passes through the points ...
 12.12.216: Prove thata b a b2a b
 12.12.217: Prove Property 6 of Theorem 8, that is,a b ca cb a bc 4
 12.12.218: Use Exercise 46 to prove that a b cb c ac a b0 a
 12.12.219: Prove that a bc d a c a d b c b d a
 12.12.220: Suppose that . (a) If a b a c, does it follow that ? b (b) If , doe...
 12.12.221: If , , and are noncoplanar vectors, let k3 v1 v2 v1 v2 v3 k2 v3 v1 ...
 12.12.222: (a) Find a vector equation and parametric equations for the line th...
 12.12.223: (a) Find parametric equations and symmetric equations of the line t...
 12.12.224: Show that the lines and with parametric equations
 12.12.225: Find an equation of the plane through the point with normal vector ...
 12.12.226: Find an equation of the plane that passes through the points , , and
 12.12.227: Find the point at which the line with parametric equations , , inte...
 12.12.228: a) Find the angle between the planes and . (b) Find symmetric equat...
 12.12.229: Find a formula for the distance from a point to the plane ax by cz ...
 12.12.230: Find the distance between the parallel planes 10x 2y 2z 5D and 5x y...
 12.12.231: In Example 3 we showed that the lines are skew. Find the distance b...
 12.12.232: Determine whether each statement is true or false. (a) Two lines pa...
 12.12.233: Find a vector equation and parametric equations for the line
 12.12.234: Find a vector equation and parametric equations for the line
 12.12.235: Find a vector equation and parametric equations for the line
 12.12.236: Find a vector equation and parametric equations for the line
 12.12.237: Find parametric equations and symmetric equations for the line.
 12.12.238: Find parametric equations and symmetric equations for the line.
 12.12.239: Find parametric equations and symmetric equations for the line.
 12.12.240: Find parametric equations and symmetric equations for the line.
 12.12.241: Find parametric equations and symmetric equations for the line.
 12.12.242: Find parametric equations and symmetric equations for the line.
 12.12.243: Find parametric equations and symmetric equations for the line.
 12.12.244: Is the line through and parallel to the line through and 5, 3, 14?
 12.12.245: Is the line through and perpendicular to the line through and 5,14?
 12.12.246: a) Find symmetric equations for the line that passes through the po...
 12.12.247: (a) Find parametric equations for the line through that is perpendi...
 12.12.248: Find a vector equation for the line segment from to 4,6,1,
 12.12.249: Find parametric equations for the line segment from to 5,6,3
 12.12.250: Determine whether the lines and are parallel, skew, or intersecting...
 12.12.251: Determine whether the lines and are parallel, skew, or intersecting...
 12.12.252: Determine whether the lines and are parallel, skew, or intersecting...
 12.12.253: Determine whether the lines and are parallel, skew, or intersecting...
 12.12.254: Find an equation of the plane.
 12.12.255: Find an equation of the plane.
 12.12.256: Find an equation of the plane.
 12.12.257: Find an equation of the plane.
 12.12.258: Find an equation of the plane.
 12.12.259: Find an equation of the plane.
 12.12.260: Find an equation of the plane.
 12.12.261: Find an equation of the plane.
 12.12.262: Find an equation of the plane.
 12.12.263: Find an equation of the plane.
 12.12.264: Find an equation of the plane.
 12.12.265: Find an equation of the plane.
 12.12.266: Find an equation of the plane.
 12.12.267: Find an equation of the plane.
 12.12.268: Find an equation of the plane.
 12.12.269: Find an equation of the plane. The plane through the origin and the...
 12.12.270: Use intercepts to help sketch the plane2x 5y z 10 3x
 12.12.271: Use intercepts to help sketch the plane3x y 2z 6 x
 12.12.272: Use intercepts to help sketch the plane6x 3y 4z 6 6
 12.12.273: Use intercepts to help sketch the plane6x 5y 3z 15
 12.12.274: Find the point at which the line intersects the given planex 3 t y ...
 12.12.275: Find the point at which the line intersects the given planex 1 2t y...
 12.12.276: Find the point at which the line intersects the given planex y 1 2z...
 12.12.277: Where does the line through and intersect the plane x y z 61
 12.12.278: Find direction numbers for the line of intersection of the planes x...
 12.12.279: Find the cosine of the angle between the planes x y z 0x and x 2y 3...
 12.12.280: Determine whether the planes are parallel, perpendicular, or neithe...
 12.12.281: Determine whether the planes are parallel, perpendicular, or neithe...
 12.12.282: Determine whether the planes are parallel, perpendicular, or neithe...
 12.12.283: Determine whether the planes are parallel, perpendicular, or neithe...
 12.12.284: Determine whether the planes are parallel, perpendicular, or neithe...
 12.12.285: Determine whether the planes are parallel, perpendicular, or neithe...
 12.12.286: (a) Find parametric equations for the line of intersection of the p...
 12.12.287: (a) Find parametric equations for the line of intersection of the p...
 12.12.288: Find symmetric equations for the line of intersection of the planes...
 12.12.289: Find symmetric equations for the line of intersection of the planes...
 12.12.290: Find an equation for the plane consisting of all points that are eq...
 12.12.291: Find an equation for the plane consisting of all points that are eq...
 12.12.292: Find an equation of the plane with intercept , intercept , and i...
 12.12.293: a) Find the point at which the given lines intersect: (b) Find an e...
 12.12.294: Find parametric equations for the line through the point that is pa...
 12.12.295: Find parametric equations for the line through the point that is pe...
 12.12.296: Which of the following four planes are parallel? Are any of them id...
 12.12.297: Use the formula in Exercise 43 in Section 12.4 to find the distance...
 12.12.298: Use the formula in Exercise 43 in Section 12.4 to find the distance...
 12.12.299: Find the distance from the point to the given plane 1, 2, 43x 2y 6z...
 12.12.300: Find the distance from the point to the given plane 6, 3, 5x 2y 4z ...
 12.12.301: Find the distance between the given parallel planes 2x 3y z 4, 4x 6...
 12.12.302: Find the distance between the given parallel planes 6z 4y 2x 9z 1 3...
 12.12.303: Show that the distance between the parallel planes and is D d1 d2 s...
 12.12.304: Find equations of the planes that are parallel to the plane and two...
 12.12.305: Show that the lines with symmetric equations and are skew, and find...
 12.12.306: Find the distance between the skew lines with parametric equations ...
 12.12.307: If , , and are not all 0, show that the equation represents a plane...
 12.12.308: Give a geometric description of each family of planes. (a) (b) (c) ...
 12.12.309: Sketch the graph of the surface z x 2 V
 12.12.310: Identify and sketch the surfaces. (a) (b)
 12.12.311: Use traces to sketch the quadric surface with equation x 2 y 2 9 z2...
 12.12.312: Use traces to sketch the surface z 4x 2 y 2
 12.12.313: Sketch the surface z y 2 x 2
 12.12.314: Sketch the surface x 2 4 y 2 z 2 4 1
 12.12.315: Identify and sketch the surface 4x 2 y 2 2z 2 4 0 S
 12.12.316: Classify the quadric surface x 2 2z2 6x y 10 0 y
 12.12.317: a) What does the equation represent as a curve in ? 2 0 (b) What do...
 12.12.318: a) Sketch the graph of as a curve in . (b) Sketch the graph of as a...
 12.12.319: Describe and sketch the surface.y 2 4z2 4
 12.12.320: Describe and sketch the surface.z 4 x
 12.12.321: Describe and sketch the surface
 12.12.322: Describe and sketch the surfaceyz 4
 12.12.323: Describe and sketch the surfacez cos x
 12.12.324: Describe and sketch the surfacex 2 y z cos x 2 1 x
 12.12.325: (a) Find and identify the traces of the quadric surface and explain...
 12.12.326: Use traces to sketch and identify the surfacex y 2 0 2 4z2 x
 12.12.327: Use traces to sketch and identify the surface9x 2 y 2 z x y 2 0 2
 12.12.328: Use traces to sketch and identify the surfacex 2 100 2 y 2 4z 2 9
 12.12.329: Use traces to sketch and identify the surface25x 2 4y 2 z x 2 100 2
 12.12.330: Use traces to sketch and identify the surfacex 2 z 0 2 4y 2 z2 4 25
 12.12.331: Use traces to sketch and identify the surface4x 2 9y x 2 z 0 2
 12.12.332: Use traces to sketch and identify the surface36x 2 16 2 y 2 36z2 36 4x
 12.12.333: Use traces to sketch and identify the surface4x 2 16y 2 z 36x 2 16
 12.12.334: Use traces to sketch and identify the surfacey z2 x
 12.12.335: Use traces to sketch and identify the surfacex y 2 z2
 12.12.336: Reduce the equation to one of the standard forms, classify the surf...
 12.12.337: Reduce the equation to one of the standard forms, classify the surf...
 12.12.338: Reduce the equation to one of the standard forms, classify the surf...
 12.12.339: Reduce the equation to one of the standard forms, classify the surf...
 12.12.340: Reduce the equation to one of the standard forms, classify the surf...
 12.12.341: Reduce the equation to one of the standard forms, classify the surf...
 12.12.342: Reduce the equation to one of the standard forms, classify the surf...
 12.12.343: Reduce the equation to one of the standard forms, classify the surf...
 12.12.344: Use a computer with threedimensional graphing software to graph th...
 12.12.345: Use a computer with threedimensional graphing software to graph th...
 12.12.346: Use a computer with threedimensional graphing software to graph th...
 12.12.347: Use a computer with threedimensional graphing software to graph th...
 12.12.348: Sketch the region bounded by the surfaces and for
 12.12.349: Sketch the region bounded by the paraboloids and
 12.12.350: Find an equation for the surface obtained by rotating the parabola ...
 12.12.351: Find an equation for the surface obtained by rotating the line abou...
 12.12.352: Find an equation for the surface consisting of all points that are ...
 12.12.353: Find an equation for the surface consisting of all points for which...
 12.12.354: Traditionally, the earths surface has been modeled as a sphere, but...
 12.12.355: A cooling tower for a nuclear reactor is to be constructed in the s...
 12.12.356: Show that if the point lies on the hyperbolic paraboloid , then the...
 12.12.357: Show that the curve of intersection of the surfaces and lies in a p...
 12.12.358: Graph the surfaces and on a common screen using the domain , and ob...
 12.12.359: What is the difference between a vector and a scalar?
 12.12.360: How do you add two vectors geometrically? How do you add them algeb...
 12.12.361: If a is a vector and c is a scalar, how is ca related to a geometri...
 12.12.362: How do you find the vector from one point to another?
 12.12.363: How do you find the dot product of two vectors if you know their le...
 12.12.364: How are dot products useful?
 12.12.365: Write expressions for the scalar and vector projections of b onto a...
 12.12.366: How do you find the cross product a b of two vectors if you know th...
 12.12.367: How are cross products useful?
 12.12.368: (a) How do you find the area of the parallelogram determined by a a...
 12.12.369: How do you find a vector perpendicular to a plane?
 12.12.370: How do you find the angle between two intersecting planes?
 12.12.371: Write a vector equation, parametric equations, and symmetric equati...
 12.12.372: Write a vector equation and a scalar equation for a plane
 12.12.373: (a) How do you tell if two vectors are parallel? (b) How do you tel...
 12.12.374: (a) Describe a method for determining whether three points , , and ...
 12.12.375: (a) How do you find the distance from a point to a line? (b) How do...
 12.12.376: What are the traces of a surface? How do you find them?
 12.12.377: Write equations in standard form of the six types of quadric surfaces.
 12.12.378: Determine whether the statement is true or false. If it is true, ex...
 12.12.379: Determine whether the statement is true or false. If it is true, ex...
 12.12.380: Determine whether the statement is true or false. If it is true, ex...
 12.12.381: Determine whether the statement is true or false. If it is true, ex...
 12.12.382: Determine whether the statement is true or false. If it is true, ex...
 12.12.383: Determine whether the statement is true or false. If it is true, ex...
 12.12.384: Determine whether the statement is true or false. If it is true, ex...
 12.12.385: Determine whether the statement is true or false. If it is true, ex...
 12.12.386: Determine whether the statement is true or false. If it is true, ex...
 12.12.387: Determine whether the statement is true or false. If it is true, ex...
 12.12.388: Determine whether the statement is true or false. If it is true, ex...
 12.12.389: Determine whether the statement is true or false. If it is true, ex...
 12.12.390: Determine whether the statement is true or false. If it is true, ex...
 12.12.391: Determine whether the statement is true or false. If it is true, ex...
 12.12.392: Determine whether the statement is true or false. If it is true, ex...
 12.12.393: Determine whether the statement is true or false. If it is true, ex...
 12.12.394: Determine whether the statement is true or false. If it is true, ex...
 12.12.395: Determine whether the statement is true or false. If it is true, ex...
 12.12.396: a) Find an equation of the sphere that passes through the point and...
 12.12.397: Copy the vectors in the figure and use them to draw each of the fol...
 12.12.398: If u and v are the vectors shown in the figure, find and . Is u v d...
 12.12.399: Calculate the given quantity if (a) (b) (c) (d) (e) (f) (g) (h) (i)...
 12.12.400: Find the values of such that the vectors and are orthogonal.
 12.12.401: Find two unit vectors that are orthogonal to both and .i 2j 3k
 12.12.402: Suppose that . Find (a) (b) (c) (d
 12.12.403: Show that if , , and are in , then a b b c c a a b c 2 c
 12.12.404: Find the acute angle between two diagonals of a cube.
 12.12.405: Given the points , , , and , find the volume of the parallelepiped ...
 12.12.406: (a) Find a vector perpendicular to the plane through the points , ,...
 12.12.407: A constant force moves an object along the line segment from to . F...
 12.12.408: A boat is pulled onto shore using two ropes, as shown in the diagra...
 12.12.409: Find the magnitude of the torque about if a 50N force is applied a...
 12.12.410: Find parametric equations for the line
 12.12.411: Find parametric equations for the line
 12.12.412: Find parametric equations for the line
 12.12.413: Find an equation of the plane.
 12.12.414: Find an equation of the plane.
 12.12.415: Find an equation of the plane.
 12.12.416: Find the point in which the line with parametric equations , , inte...
 12.12.417: Find the distance from the origin to the line x 1 t y 2 t z 1 2t 2x y
 12.12.418: Determine whether the lines given by the symmetric equations and ar...
 12.12.419: (a) Show that the planes and 2x 3y 4z 5 are neither parallel nor pe...
 12.12.420: (b) Find, correct to the nearest degree, the angle between these pl...
 12.12.421: Find an equation of the plane through the line of intersection of t...
 12.12.422: (a) Find an equation of the plane that passes through the points , ...
 12.12.423: Find the distance between the planes and 3x y 4z 24
 12.12.424: Identify and sketch the graph of each surface x 3
 12.12.425: Identify and sketch the graph of each surface x z
 12.12.426: Identify and sketch the graph of each surface y z2
 12.12.427: Identify and sketch the graph of each surface x 2 y 2 4z2 y
 12.12.428: Identify and sketch the graph of each surface 4x y 2z 4 2
 12.12.429: Identify and sketch the graph of each surface 4x 2 y 2 4z 4x y 2z 4...
 12.12.430: Identify and sketch the graph of each surface y 2 z2 1 x 2 4
 12.12.431: Identify and sketch the graph of each surface 4x 2 4y 2 8y z2 0 y
 12.12.432: Identify and sketch the graph of each surface x y2 z2 2y 4z 5 4x
 12.12.433: An ellipsoid is created by rotating the ellipse about the axis. Fi...
 12.12.434: A surface consists of all points such that the distance from to the...
 12.12.435: Each edge of a cubical box has length 1 m. The box contains nine sp...
 12.12.436: Let be a solid box with length , width , and height . Let be the se...
 12.12.437: Let be the line of intersection of the planes and , where is a real...
 12.12.438: A plane is capable of flying at a speed of 180 kmh in still air. Th...
 12.12.439: Suppose a block of mass is placed on an inclined plane, as shown in...
Solutions for Chapter 12: VECTORS AND THE GEOMETRY OF SPACE
Full solutions for Calculus: Early Transcendentals  6th Edition
ISBN: 9780495011668
Solutions for Chapter 12: VECTORS AND THE GEOMETRY OF SPACE
Get Full SolutionsThis textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 6. Since 439 problems in chapter 12: VECTORS AND THE GEOMETRY OF SPACE have been answered, more than 18941 students have viewed full stepbystep solutions from this chapter. Calculus: Early Transcendentals was written by Sieva Kozinsky and is associated to the ISBN: 9780495011668. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 12: VECTORS AND THE GEOMETRY OF SPACE includes 439 full stepbystep solutions.

Binomial
A polynomial with exactly two terms

Binomial theorem
A theorem that gives an expansion formula for (a + b)n

Branches
The two separate curves that make up a hyperbola

Convenience sample
A sample that sacrifices randomness for convenience

Convergence of a series
A series aqk=1 ak converges to a sum S if imn: q ank=1ak = S

Difference of functions
(ƒ  g)(x) = ƒ(x)  g(x)

Equivalent systems of equations
Systems of equations that have the same solution.

Expanded form of a series
A series written explicitly as a sum of terms (not in summation notation).

Hyperboloid of revolution
A surface generated by rotating a hyperbola about its transverse axis, p. 607.

Index of summation
See Summation notation.

Linear regression
A procedure for finding the straight line that is the best fit for the data

Quadratic regression
A procedure for fitting a quadratic function to a set of data.

Random variable
A function that assigns realnumber values to the outcomes in a sample space.

Sequence of partial sums
The sequence {Sn} , where Sn is the nth partial sum of the series, that is, the sum of the first n terms of the series.

Sinusoid
A function that can be written in the form f(x) = a sin (b (x  h)) + k or f(x) = a cos (b(x  h)) + k. The number a is the amplitude, and the number h is the phase shift.

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

Stemplot (or stemandleaf plot)
An arrangement of a numerical data set into a specific tabular format.

Transformation
A function that maps real numbers to real numbers.

Upper bound for real zeros
A number d is an upper bound for the set of real zeros of ƒ if ƒ(x) ? 0 whenever x > d.

Weighted mean
A mean calculated in such a way that some elements of the data set have higher weights (that is, are counted more strongly in determining the mean) than others.
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