Chapter 7

Determine whether the statement is true or false. Justify your answer. The work \(W\) done by a constant force \(\mathbf{F}\) acting along the line of motion of an object is represented by a vector.

If \(\mathbf{u}=\langle\cos \theta, \sin \theta\rangle\) and \(\mathbf{v}=\langle\sin \theta,-\cos \theta\rangle,\) are \(\mathbf{u}\) and \(\mathbf{v}\) orthogonal, parallel, or neither? Explain.

Error Analysis Describe the error. \(\langle5,8\rangle \cdot\langle=2,7\rangle=\langle-10,56\rangle\)

Let \(\mathbf{u}\) be a unit vector. What is the value of \(\mathbf{u} \cdot \mathbf{u} ?\) Explain.

What is known about \(\theta,\) the angle between two nonzero vectors \(\mathbf{u}\) and \(\mathbf{v}\) (see figure) under each condition? (a) \(\mathbf{u} \cdot \mathbf{v}=0\) (b) \(\mathbf{u} \cdot \mathbf{v}>0\) (c) \(\mathbf{u} \cdot \mathbf{v}<0\)

Find the component form of v given its magnitude and the angle it makes with the positive \(x\) -axis. Sketch v. Angle:$$\begin{aligned} &\theta=0^{\circ}\\\ &\theta=45^{\circ}\\\ &\theta=120^{\circ}\\\ &\theta=135^{\circ}\\\ &\theta=150^{\circ}\\\ &\theta=90^{\circ}\\\ &\mathbf{v} \text { in the direction } \mathbf{i}+3 \mathbf{j}\\\ &\mathbf{v} \text { in the direction } 3 \mathbf{i}+4 \mathbf{j} \end{aligned}$$ Magnitude:$$\|\mathbf{v}\|=4 \sqrt{3}$$

What can be said about the vectors \(\mathbf{u}\) and \(\mathbf{v}\) under each condition? (a) The projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{u}.\) (b) The projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(0 .\)

Use vectors to prove that the diagonals of a rhombus are perpendicular.

Prove the following. $$\|\mathbf{u}-\mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2}-2 \mathbf{u} \cdot \mathbf{v}$$

Prove that if \(\mathbf{u}\) is orthogonal to \(\mathbf{v}\) and \(\mathbf{w},\) then \(\mathbf{u}\) is orthogonal to \(c \mathbf{v}+d \mathbf{w}\) for any scalars \(c\) and \(d .\)

Prove that if \(\mathbf{u}\) is a unit vector and \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{i},\) then \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}.\)

Prove that if \(\mathbf{u}\) is a unit vector and \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{j},\) then $$\mathbf{u}=\cos \left(\frac{\pi}{2}-\theta\right) \mathbf{i}+\sin \left(\frac{\pi}{2}-\theta\right) \mathbf{j}.$$

Use the Law of cosines to find the angle \(\alpha\) between the vectors. (Assume \(0^{\circ} \leq \alpha \leq 180^{\circ}\) ). $$\mathbf{v}=\mathbf{i}+\mathbf{j}, \quad \mathbf{w}=2(\mathbf{i}-\mathbf{j})$$

Describe how the graph of \(g\) is related to the graph of \(f.\) $$g(x)=f(x-4)$$

Describe how the graph of \(g\) is related to the graph of \(f.\) $$g(x)=-f(x)$$

Use the Law of cosines to find the angle \(\alpha\) between the vectors. (Assume \(0^{\circ} \leq \alpha \leq 180^{\circ}\) ). $$\mathbf{v}=3 \mathbf{i}+\mathbf{j}, \quad \mathbf{w}=2 \mathbf{i}-\mathbf{j}$$

Represent the powers \(z, z^{2}, z^{3},\) and \(z^{4}\) graphically. Describe the pattern. $$z=\frac{\sqrt{2}}{2}(1+i)$$

Describe how the graph of \(g\) is related to the graph of \(f.\) $$g(x)=f(x)+6$$

Represent the powers \(z, z^{2}, z^{3},\) and \(z^{4}\) graphically. Describe the pattern. $$z=\frac{1}{2}(1+\sqrt{3} i)$$

Describe how the graph of \(g\) is related to the graph of \(f.\) $$g(x)=f(2 x)$$

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$|z|=3$$

Find the angle between the forces given the magnitude of their resultant. (Hint: Write force 1 as a vector in the direction of the positive \(x\) -axis and force 2 as a vector at an angle \(\theta\) with the positive \(x\) -axis.). Force 1:45 pounds Force 2:60 pounds Resultant Force:90 pounds

Perform the operation and write the result in standard form. $$3 i(4-5 i)$$

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$|z|=5$$

Perform the operation and write the result in standard form. $$-2 i(1+6 i)$$

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$|z|=4$$

Perform the operation and write the result in standard form. $$(1+3 i)(1-3 i)$$

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$|z|=6$$

A gun with a muzzle velocity of 1200 feet per second is fired at an angle of \(6^{\circ}\) above the horizontal. Find the vertical and horizontal components of the velocity.

Perform the operation and write the result in standard form. $$(7-4 i)(7+4 i)$$

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$|z|=7$$

Perform the operation and write the result in standard form. $$\frac{3}{1+i}+\frac{2}{2-3 i}$$

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$|z|=8$$

Perform the operation and write the result in standard form. $$\frac{6}{4-i}-\frac{3}{1+i}$$

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$\theta=\frac{\pi}{6}$$

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$\theta=\frac{\pi}{4}$$

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$\theta=\frac{\pi}{3}$$

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$\theta=\frac{\pi}{2}$$

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$\theta=\frac{2 \pi}{3}$$

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$\theta=\frac{3 \pi}{4}$$

A commercial jet is flying from Miami to Seattle. The jet's velocity with respect to the air is 580 miles per hour, and its bearing is \(332^{\circ} .\) The wind, at the altitude of the jet, is blowing from the southwest with a velocity of 60 miles per hour. (a) Draw a figure that gives a visual representation of the problem. (b) Write the velocity of the wind as a vector in component form. (c) Write the velocity of the jet relative to the air as a vector in component form. (d) What is the speed of the jet with respect to the ground? (e) What is the true direction of the jet?

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$(1+i)^{3}$$

Determine whether the statement is true or false. Justify your answer.If \(\mathbf{u}\) and \(\mathbf{v}\) have the same magnitude and direction, then \(\mathbf{u}=\mathbf{v}\).

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$(2+2 i)^{6}$$

Determine whether the statement is true or false. Justify your answer.If \(\mathbf{u}\) is a unit vector in the direction of \(\mathbf{v},\) then \(\mathbf{v}=\|\mathbf{v}\| \mathbf{u}\).

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$(-1+i)^{6}$$

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$(3-3 i)^{8}$$

Determine whether the statement is true or false. Justify your answer.If \(\mathbf{u}=a \mathbf{i}+b \mathbf{j}\) is a unit vector, then \(a^{2}+b^{2}=1\).

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$2(\sqrt{3}-i)^{5}$$

Consider two forces of equal magnitude acting on a point. (a) If the magnitude of the resultant is the sum of the magnitudes of the two forces, make a conjecture about the angle between the forces. (b) If the resultant of the forces is \(0,\) make a conjecture about the angle between the forces. (c) Can the magnitude of the resultant be greater than the sum of the magnitudes of the two forces? Explain.