?(a) Find an equation of the plane that passes through the points , and .(b) Find

What is the difference between a vector and a scalar?

How do you add two vectors geometrically? How do you add them algebraically?

If a is a vector and c is a scalar, how is ca related to a geometrically? How do you find ca algebraically?

How do you find the vector from one point to another?

How do you find the dot product a b of two vectors if you know their lengths and the angle between them? What if you know their components?

How are dot products useful?

Write expressions for the scalar and vector projections of b onto a. Illustrate with diagrams.

How do you find the cross product a b of two vectors if you know their lengths and the angle between them? What if you know their components?

How are cross products useful?

(a) How do you find the area of the parallelogram determined by a and b? (b) How do you find the volume of the parallelepiped determined by a, b, and c?

How do you find a vector perpendicular to a plane?

How do you find the angle between two intersecting planes?

Write a vector equation, parametric equations, and symmetric equations for a line.

Write a vector equation and a scalar equation for a plane.

(a) How do you tell if two vectors are parallel? (b) How do you tell if two vectors are perpendicular? (c) How do you tell if two planes are parallel?

(a) Describe a method for determining whether three points P, Q, and R lie on the same line. (b) Describe a method for determining whether four points P, Q, R, and S lie in the same plane.

(a) How do you find the distance from a point to a line? (b) How do you find the distance from a point to a plane? (c) How do you find the distance between two lines?

What are the traces of a surface? How do you find them?

Write equations in standard form of the six types of quadric surfaces.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(\mathbf{u}=\left\langle u_1, u_2\right\rangle\) and \(\mathbf{v}=\left\langle v_1, v_2\right\rangle\), then \(\mathbf{u} \cdot \mathbf{v}=\left\langle u_1 v_1, u_2 v_2\right\rangle\).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. For any vectors and in .

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. For any vectors \(\mathbf{u}\) and \(\mathbf{v}\) in \(V_3,|\mathbf{u} \cdot \mathbf{v}|=|\mathbf{u}||\mathbf{v}|\).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. For any vectors u and v in \(V_{3}\), \(|\mathbf{u} \times \mathbf{v}|=|\mathbf{u}||\mathbf{v}|\)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. For any vectors \(\mathbf{u}\) and \(\mathbf{v}\) in \(V_3, \mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}\).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. For any vectors and in

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. For any vectors and in

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. For any vectors and in and any scalar ,

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. For any vectors u and v in \(V_{3}\) and any scalar k, \(k(\mathbf{u} \times \mathbf{v})=(k \mathbf{u}) \times \mathbf{v}\)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. For any vectors , and in ,

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. For any vectors , and in ,

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. For any vectors u, v, and w in \(V_{3}\), \(\mathbf{u} \times(\mathbf{v} \times \mathbf{w})=(\mathbf{u} \times \mathbf{v}) \times \mathbf{w}\)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. For any vectors and in .

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. For any vectors and in

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The vector is parallel to the plane

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. A linear equation represents a line in space.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The set of points is a circle.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. In the graph of is a paraboloid.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(\mathbf{u} \cdot \mathbf{v}=0\), then \(\mathbf{u}=\mathbf{0}\) or \(\mathbf{v}=\mathbf{0}\).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If , then or .

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If and , then or .

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(\mathbf{u}\) and \(\mathbf{v}\) are in \(V_3\), then \(|\mathbf{u} \cdot \mathbf{v}| \leqslant|\mathbf{u}||\mathbf{v}|\).

(a) Find an equation of the sphere that passes through the point (6, -2, 3) and has center (-1, 2, 1) (b) Find the curve in which this sphere intersects the yz-plane. (c) Find the center and radius of the sphere \(x^{2}+y^{2}+z^{2}-8 x+2 y+6 z+1=0\)

Copy the vectors in the figure and use them to draw each of the following vectors. (a) (b) (c) (d)

If and are the vectors shown in the figure, find and Is directed into the page or out of it?

Calculate the given quantity if \(\begin{aligned}&\mathbf{a}=\mathbf{i}+\mathbf{j}-\mathbf{2} \mathbf{k} \\&\mathbf{b}=3\mathbf{i}-2 \mathbf{j}+\mathbf{k} \\&\mathrm{c}=\mathbf{j}-5 \mathrm{k}\end{aligned}\) (a) \(2 \mathbf{a}+3 \mathbf{b}\) (b) \(|\mathbf{b}|\) (c) \(\mathbf{a} \cdot \mathbf{b}\) (d) \(\mathbf{a} \times \mathbf{b}\) (e) \(|\mathbf{b} \times \mathbf{c}|\) (f) \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})\) (g) \(\mathbf{c} \times \mathbf{c}\) (h) v\mathbf{a} \times(\mathbf{b} \times \mathbf{c})\) (i) \(\operatorname{comp}_a \mathbf{b}\) (j) \(\operatorname{proj}_{\mathbf{a}} \mathbf{b}\) (k) The angle between \(\mathbf{a}\) and\(\mathbf{b}\) (correct to the nearest degree)

Find the values of such that the vectors and are orthogonal.

Find two unit vectors that are orthogonal to both and .

Suppose that . Find the value of each of the following. (a) (b) (c) (d)

Show that if \(\mathbf{a}, \mathbf{b}\), and \(\mathbf{c}\) are in \(V_3\), then \((\mathbf{a} \times \mathbf{b}) \cdot[(\mathbf{b} \times \mathbf{c}) \times(\mathbf{c} \times \mathbf{a})]=[\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})]^2\)

Find the acute angle between two diagonals of a cube.

Given the points , and , find the volume of the parallelepiped with adjacent edges , and .

(a) Find a vector perpendicular to the plane through the points , and . (b) Find the area of triangle .

A constant force moves an object along the line segment from to . Find the work done if the distance is measured in meters and the force in newtons.

A boat is pulled onto shore using two ropes, as shown in the diagram. If a force of is needed, find the magnitude of the force in each rope.

Find the magnitude of the torque about P if a 50-N force is applied as shown.

Find parametric equations for the line. The line through and

Find parametric equations for the line. The line through and parallel to the line

Find parametric equations for the line. The line through and perpendicular to the plane

Find an equation of the plane. The plane through and parallel to

Find an equation of the plane. The plane through , and

Find an equation of the plane. The plane through (1, 2, -2) that contains the line x = 2t, y = 3 - t, z = 1 + 3t

Find the point in which the line with parametric equations intersects the plane

Find the distance from the origin to the line

Determine whether the lines given by the symmetric equations and are parallel, skew, or intersecting.

(a) Show that the planes and are neither parallel nor perpendicular. (b) Find, correct to the nearest degree, the angle between these planes.

Find an equation of the plane through the line of intersection of the planes x - z = 1 and y + 2 z = 3 and perpendicular to the plane x + y - 2z = 1.

(a) Find an equation of the plane that passes through the points , and . (b) Find symmetric equations for the line through that is perpendicular to the plane in part (a). (c) A second plane passes through and has normal vector Show that the acute angle between the planes is approximately . (d) Find parametric equations for the line of intersection of the two planes.

Find the distance between the planes and

Identify and sketch the graph of each surface.

Identify and sketch the graph of each surface.

Identify and sketch the graph of each surface.

Identify and sketch the graph of each surface. \(x^{2}=y^{2}+4 z^{2}\)

Identify and sketch the graph of each surface.

Identify and sketch the graph of each surface.

Identify and sketch the graph of each surface.

Identify and sketch the graph of each surface.

Identify and sketch the graph of each surface. \(x=y^{2}+z^{2}-2 y-4 z+5\)

An ellipsoid is created by rotating the ellipse about the -axis. Find an equation of the ellipsoid.

A surface consists of all points such that the distance from to the plane is twice the distance from to the point . Find an equation for this surface and identify it.

Each edge of a cubical box has length . The box contains nine spherical balls with the same radius . The center of one ball is at the center of the cube and it touches the other eight balls. Each of the other eight balls touches three sides of the box. Thus the balls are tightly packed in the box (see the figure). Find . (If you have trouble with this problem, read about the problem-solving strategy entitled Use Analogy in Principles of Problem Solving following Chapter )

Let be a solid box with length , width , and height . Let be the set of all points that are a distance at most 1 from some point of . Express the volume of in terms of , and .

Let be the line of intersection of the planes and , where is a real number. (a) Find symmetric equations for . (b) As the number varies, the line sweeps out a surface . Find an equation for the curve of intersection of with the horizontal plane (the trace of in the plane ). (c) Find the volume of the solid bounded by and the planes and .

A plane is capable of flying at a speed of \(180 \mathrm{~km} / \mathrm{h}\) in still air. The pilot takes off from an airfield and heads due north according to the plane's compass. After 30 minutes of flight time, the pilot notices that, due to the wind, the plane has actually traveled \(80 \mathrm{~km}\) in the direction \(\mathrm{N} 5^{\circ} \mathrm{E}\). (a) What is the wind velocity? (b) In what direction should the pilot have headed to reach the intended destination?

Suppose \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) are vectors with \(\left|\mathbf{v}_{1}\right|=2,\left|\mathbf{v}_{2}\right|=3\), and \(\mathbf{v}_{1} \cdot \mathbf{v}_{2}=5\). Let \(\mathbf{v}_{3}=\operatorname{proj}_{\mathbf{v}_{1}} \mathbf{v}_{2}\), \(\mathbf{v}_{4}=\operatorname{proj}_{\mathbf{v}_{2}} \mathbf{v}_{3}, \mathbf{v}_{5}=\operatorname{proj}_{\mathbf{v}_{3}} \mathbf{v}_{4}\), and so on. Compute \(\Sigma_{n=1}^{\infty}\left|\mathbf{v}_{n}\right|\). Equation Transcription: ????1 ????2 | ????1 | = 2, | ????2 | =3 ????1 ????2 =5 ????3 = projv1 ????2 ????4 = projv2 ????3 ????5 = projv3 ????4 | ????n | Text Transcription: v_1 v_2 | v_1 | = 2, | v_2 | =3 v_1 cdot v_2 =5 v_3 = proj_v1 v_2 v_4 = proj_v2 v_3 v_5 = proj_v3 v_4 sum _n=1 ^infty | v_n |

Find an equation of the largest sphere that passes through the point and is such that each of the points inside the sphere satisfies the condition

Suppose a block of mass m is placed on an inclined plane, as shown in the figure. The block's descent down the plane is slowed by friction; if \(\theta\) is not too large, friction will prevent the block from moving at all. The forces acting on the block are the weight W, where | W | = mg (g is the acceleration due to gravity); the normal force \(\mathbf{N}\) (the normal component of the reactionary force of the plane on the block), where | N | = n; and the force F due to friction, which acts parallel to the inclined plane, opposing the direction of motion. If the block is at rest and \(\theta\) is increased, | F | must also increase until ultimately | F | reaches its maximum, beyond which the block begins to slide. At this angle \(\theta_{s}\) it has been observed that | F | is proportional to n. Thus, when | F | is maximal, we can say that \(|\mathbf{F}|=\mu_{s} n\), where \(\mu_{s}\) is called the coefficient of static friction and depends on the materials that are in contact. (a) Observe that N + F + W = 0 and deduce that \(\mu_{s}=\tan \theta_{s}\). (b) Suppose that, for \(\theta>\theta_{s}\), an additional outside force H is applied to the block, horizontally from the left, and let | H | = h. If h is small, the block may still slide down the plane; if h is large enough, the block will move up the plane. Let \(h_{\min }\) be the smallest value of h that allows the block to remain motionless (so that | F | is maximal). By choosing the coordinate axes so that F lies along the x-axis, resolve each force into components parallel and perpendicular to the inclined plane and show that \(h_{\min } \sin \ \theta+m g \cos \ \theta=n\) and \(h_{\min } \cos \ \theta+\mu_{s} n=m g \sin \ \theta\) (c) Show that \(h_{\min }=m g \tan \left(\theta-\theta_{s}\right)\) Does this equation seem reasonable? Does it make sense for \(\theta=\theta_{s}\) Does it make sense as \(\theta \rightarrow 90^{\circ}\)? Explain. (d) Let \(h_{m a x}\) be the largest value of h that allows the block to remain motionless. (In which direction is F heading?) Show that \(h_{\max }=m g \tan \left(\theta+\theta_{s}\right)\) Does this equation seem reasonable? Explain.

A solid has the following properties. When illuminated by rays parallel to the -axis, its shadow is a circular disk. If the rays are parallel to the -axis, its shadow is a square. If the rays are parallel to the -axis, its shadow is an isosceles triangle. (In Exercise 12.1.52 you were asked to describe and sketch an example of such a solid, but there are many such solids.) Assume that the projection onto the -plane is a square whose sides have length 1 . (a) What is the volume of the largest such solid? (b) Is there a smallest volume?