Chapter 10

Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be vectors, and let \(a\) be a scalar. Prove the given property. $$ (\mathbf{u}-\mathbf{v}) \cdot(\mathbf{u}+\mathbf{v})=|\mathbf{u}|^{2}-|\mathbf{v}|^{2} $$

\(41-46\) . Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors \(\mathbf{i}\) and \(\mathbf{j}\) . $$ |\mathbf{v}|=50, \quad \theta=120^{\circ} $$

Two direction angles of a vector are given. Find the third direction angle, given that it is either obtuse or acute as indicated. (In Exercises 43 and 44, round your answers to the nearest degree.) $$ \alpha=60^{\circ}, \quad \beta=50^{\circ} ; \quad \gamma \text { is obtuse } $$

\(41-46\) . Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors \(\mathbf{i}\) and \(\mathbf{j}\) . $$ |\mathbf{v}|=1, \quad \theta=225^{\circ} $$

\(41-46\) . Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors \(\mathbf{i}\) and \(\mathbf{j}\) . $$ |\mathbf{v}|=800, \quad \theta=125^{\circ} $$

Explain why it is impossible for a vector to have the given direction angles. $$ \alpha=20^{\circ}, \quad \beta=45^{\circ} $$

Work The force \(\mathbf{F}=4 \mathbf{i}-7 \mathbf{j}\) moves an object 4 \(\mathrm{ft}\) along the \(x\) -axis in the positive direction. Find the work done if the unit of force is the pound.

\(41-46\) . Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors \(\mathbf{i}\) and \(\mathbf{j}\) . $$ |\mathbf{v}|=4, \quad \theta=10^{\circ} $$

Explain why it is impossible for a vector to have the given direction angles. $$ \alpha=150^{\circ}, \quad \gamma=25^{\circ} $$

Work A constant force \(\mathbf{F}=\langle 2,8\rangle\) moves an object along a straight line from the point \((2,5)\) to the point \((11,13) .\) Find the work done if the distance is measured in feet and the force is measured in pounds.

\(41-46\) . Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors \(\mathbf{i}\) and \(\mathbf{j}\) . $$ |\mathbf{v}|=\sqrt{3}, \quad \theta=300^{\circ} $$

Resultant of Four Forces An object located at the origin in a three- dimensional coordinate system is held in equilibrium by four forces. One has magnitude 7 lb and points in the direction of the positive x-axis, so it is represented by the vector 7i. The second has magnitude 24 lb and points in the direction of the positive y-axis. The third has magnitude 25 lb and points in the direction of the negative z-axis. (a) Use the fact that the four forces are in equilibrium (that is, their sum is 0) to find the fourth force. Express it in terms of the unit vectors i, j, and k. (b) What is the magnitude of the fourth force?

Work A lawn mower is pushed a distance of 200 \(\mathrm{ft}\) along a horizontal path by a constant force of 50 lb. The handle of the lawn mower is held at an angle of \(30^{\circ}\) from the horizontal (see the figure). Find the work done.

\(47-52\) . Find the magnitude and direction (in degrees) of the vector. $$ \mathbf{v}=\langle 3,4\rangle $$

Central Angle of a Tetrahedron A tetrahedron is a solid with four triangular faces, four vertices, and six edges, as shown in the figure. In a regular tetrahedron, the edges are all of the same length. Consider the tetrahedron with vertices \(A(1,0,0), B(0,1,0), C(0,0,1),\) and \(D(1,1,1)\) (a) Show that the tetrahedron is regular. (b) The center of the tetrahedron is the point \(E\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)\) (the "average" of the vertices). Find the angle between the vectors that join the center to any two of the vertices (for instance, \(\angle A E B\) ). This angle is called the central angle of the tetrahedron.

Work A car drives 500 \(\mathrm{ft}\) on a road that is inclined \(12^{\circ}\) to the horizontal, as shown in the following figure. The car weighs 2500 Ib. Thus gravity acts straight down on the car with a constant force \(\mathbf{F}=-2500 \mathbf{j}\) . Find the work done by the car in overcoming gravity.

\(47-52\) . Find the magnitude and direction (in degrees) of the vector. $$ \mathbf{v}=\left\langle-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right\rangle $$

Parallel Vectors Two nonzero vectors are parallel if they point in the same direction or in opposite directions. This means that if two vectors are parallel, one must be a scalar multiple of the other. Determine whether the given vectors \(\mathbf{u}\) and \(\mathbf{v}\) are parallel. If they are, express \(\mathbf{v}\) as a scalar multiple of \(\mathbf{u} .\) (a) \(\mathbf{u}=\langle 3,-2,4\rangle, \mathbf{v}=\langle- 6,4,-8\rangle\) (b) \(\mathbf{u}=\langle- 9,-6,12\rangle, \mathbf{v}=\langle 12,8,-16\rangle\) (c) \(\mathbf{u}=\mathbf{i}+\mathbf{j}+\mathbf{k}, \quad \mathbf{v}=2 \mathbf{i}+2 \mathbf{j}-2 \mathbf{k}\)

Force \(A\) car is on a driveway that is inclined \(25^{\circ}\) to the horizontal. If the car weighs 2755 Ib, find the force required to keep it from rolling down the driveway.

\(47-52\) . Find the magnitude and direction (in degrees) of the vector. $$ \mathbf{v}=\langle- 12,5\rangle $$

Unit Vectors A unit vector is a vector of magnitude 1. Multiplying a vector by a scalar changes its magnitude but not its direction. (a) If a vector v has magnitude \(m,\) what scalar multiple of \(\mathbf{v}\) has magnitude 1 (i.e., is a unit vector)? (b) Multiply each of the following vectors by an appropriate scalar to change them into unit vectors: $$ \langle 1,-2,2\rangle \quad\langle- 6,8,-10\rangle \quad\langle 6,5,9\rangle $$

Force \(A\) car is on a driveway that is inclined \(10^{\circ}\) to the horizontal. A force of 490 lb is required to keep the car from rolling down the driveway. (a) Find the weight of the car. (b) Find the force the car exerts against the driveway.

\(47-52\) . Find the magnitude and direction (in degrees) of the vector. $$ \mathbf{v}=\langle 40,9\rangle $$

Vector Equation of a Sphere Let \(a=\langle 2,2,2\rangle\) \(\mathbf{b}=\langle- 2,-2,0\rangle,\) and \(\mathbf{r}=\langle x, y, z\rangle .\) (a) Show that the vector equation \((\mathbf{r}-\mathbf{a}) \cdot(\mathbf{r}-\mathbf{b})=0\) rep- resents a sphere, by expanding the dot product and simplifying the resulting algebraic equation. (b) Find the center and radius of the sphere. (c) Interpret the result of part (a) geometrically, using the fact that the dot product of two vectors is 0 only if the vectors are perpendicular. \([\text { Hint: Draw a diagram showing the }\) endpoints of the vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{r},\) noting that the end- points of \(\mathbf{a}\) and \(\mathbf{b}\) are the endpoints of a diameter and the endpoint of \(\mathbf{r}\) is an arbitrary point on the sphere. \(]\) (d) Using your observations from part (a), find a vector equation for the sphere in which the points \((0,1,3)\) and \((2,-1,4)\) form the endpoints of a diameter. Simplify the vector equation to obtain an algebraic equation for the sphere. What are its center and radius?

Force A package that weighs 200 lb is placed on an inclined plane. If a force of 80 lb is just sufficient to keep the package from sliding, find the angle of inclination of the plane. (Ignore the effects of friction.)

\(47-52\) . Find the magnitude and direction (in degrees) of the vector. $$ \mathbf{v}=\mathbf{i}+\sqrt{3} \mathbf{j} $$

\(47-52\) . Find the magnitude and direction (in degrees) of the vector. $$ \mathbf{v}=\mathbf{i}+\mathbf{j} $$

Components of a Force A man pushes a lawn mower with a force of 30 lb exerted at an angle of \(30^{\circ}\) to the ground. Find the horizontal and vertical components of the force.

Components of a Velocity A jet is flying in a direction \(\mathrm{N} 20^{\circ} \mathrm{E}\) with a speed of 500 \(\mathrm{mi} / \mathrm{h}\) . Find the north and east components of the velocity.

Velocity A river flows due south at 3 milh. A swimmer attempting to cross the river heads due east swimming at 2 milh relative to the water. Find the true velocity of the swimmer as a vector.

Velocity The speed of an airplane is 300 \(\mathrm{mi} / \mathrm{h}\) relative to the air. The wind is blowing due north with a speed of 30 \(\mathrm{mi} / \mathrm{h}\) . In what direction should the airplane head in order to arrive at a point due west of its location?

Velocity A migrating salmon heads in the direction \(\mathrm{N} 45^{\circ} \mathrm{E}\) , swimming at 5 \(\mathrm{mi} / \mathrm{h}\) relative to the water. The prevailing ocean currents flow due east at 3 \(\mathrm{mi} / \mathrm{h}\) . Find the true velocity of the fish as a vector.

True Velocity of a Jet A pilot heads his jet due east. The jet has a speed of 425 milh relative to the air. The wind is blowing due north with a speed of 40 \(\mathrm{mi} / \mathrm{h}\) . (a) Express the velocity of the wind as a vector in component form. (b) Express the velocity of the jet relative to the air as a vector in component form. (c) Find the true velocity of the jet as a vector. (d) Find the true speed and direction of the jet.

True Velocity of a Jet A jet is flying through a wind that is blowing with a speed of 55 \(\mathrm{mi} / \mathrm{h}\) in the direction \(\mathrm{N} 30^{\circ} \mathrm{E}\) (see the figure). The jet has a speed of 765 \(\mathrm{mi} / \mathrm{h}\) relative to the air, and the pilot heads the jet in the direction \(\mathrm{N} 45^{\circ} \mathrm{E}\) (a) Express the velocity of the wind as a vector in component form. (b) Express the velocity of the jet relative to the air as a vector in component form. (c) Find the true velocity of the jet as a vector. (d) Find the true speed and direction of the jet.

Velocity of a Boat A straight river flows east at a speed of 10 \(\mathrm{mi} / \mathrm{h}\) . A boater starts at the south shore of the river and heads in a direction \(60^{\circ}\) from the shore (see the figure). The motorboat has a speed of 20 \(\mathrm{mi} / \mathrm{h}\) relative to the water. (a) Express the velocity of the river as a vector in component form. (b) Express the velocity of the motorboat relative to the water as a vector in component form. (c) Find the true velocity of the motorboat. (d) Find the true speed and direction of the motorboat.

Velocity of a Boat A boat heads in the direction \(\mathrm{N} 72^{\circ} \mathrm{E}\) The speed of the boat relative to the water is 24 \(\mathrm{mi} / \mathrm{h}\) . The water is flowing directly south. It is observed that the true direction of the boat is directly east. (a) Express the velocity of the boat relative to the water as a vector in component form. (b) Find the speed of the water and the true speed of the boat.

Velocity A woman walks due west on the deck of an ocean liner at 2 milh. The ocean liner is moving due north at a speed of 25 \(\mathrm{mi} / \mathrm{h}\) . Find the speed and direction of the woman relative to the surface of the water.

\(67-72\) . Equilibrium of Forces The forces \(\mathbf{F}_{1}, \mathbf{F}_{2}, \ldots, \mathbf{F}_{n}\) acting at the same point \(P\) are said to be in equilibrium if the resultant force is zero, that is, if \(\mathbf{F}_{1}+\mathbf{F}_{2}+\cdots+\mathbf{F}_{n}=0 .\) Find (a) the resultant forces acting at \(P,\) and (b) the additional force required (if any) for the forces to be in equilibrium. $$ \mathbf{F}_{1}=\langle 2,5\rangle, \quad \mathbf{F}_{2}=\langle 3,-8\rangle $$

\(67-72\) . Equilibrium of Forces The forces \(\mathbf{F}_{1}, \mathbf{F}_{2}, \ldots, \mathbf{F}_{n}\) acting at the same point \(P\) are said to be in equilibrium if the resultant force is zero, that is, if \(\mathbf{F}_{1}+\mathbf{F}_{2}+\cdots+\mathbf{F}_{n}=0 .\) Find (a) the resultant forces acting at \(P,\) and (b) the additional force required (if any) for the forces to be in equilibrium. $$ \mathbf{F}_{1}=\langle 3,-7\rangle, \quad \mathbf{F}_{2}=\langle 4,-2\rangle, \quad \mathbf{F}_{3}=\langle- 7,9\rangle $$

\(67-72\) . Equilibrium of Forces The forces \(\mathbf{F}_{1}, \mathbf{F}_{2}, \ldots, \mathbf{F}_{n}\) acting at the same point \(P\) are said to be in equilibrium if the resultant force is zero, that is, if \(\mathbf{F}_{1}+\mathbf{F}_{2}+\cdots+\mathbf{F}_{n}=0 .\) Find (a) the resultant forces acting at \(P,\) and (b) the additional force required (if any) for the forces to be in equilibrium. $$ \begin{array}{l}{\mathbf{F}_{1}=4 \mathbf{i}-\mathbf{j}, \quad \mathbf{F}_{2}=3 \mathbf{i}-7 \mathbf{j}, \quad \mathbf{F}_{3}=-8 \mathbf{i}+3 \mathbf{j}} \\ {\mathbf{F}_{4}=\mathbf{i}+\mathbf{j}}\end{array} $$

\(67-72\) . Equilibrium of Forces The forces \(\mathbf{F}_{1}, \mathbf{F}_{2}, \ldots, \mathbf{F}_{n}\) acting at the same point \(P\) are said to be in equilibrium if the resultant force is zero, that is, if \(\mathbf{F}_{1}+\mathbf{F}_{2}+\cdots+\mathbf{F}_{n}=0 .\) Find (a) the resultant forces acting at \(P,\) and (b) the additional force required (if any) for the forces to be in equilibrium. $$ \mathbf{F}_{1}=\mathbf{i}-\mathbf{j}, \quad \mathbf{F}_{2}=\mathbf{i}+\mathbf{j}, \quad \mathbf{F}_{3}=-2 \mathbf{i}+\mathbf{j} $$

Vectors That Form a Polygon Suppose that \(n\) vectors can be placed head to tail in the plane so that they form a polygon. (The figure shows the case of a hexagon.) Explain why the sum of these vectors is 0 .