Chapter 10

Answers are given at the end of these exercises. The conjugate of \(-4-3 i\) is _______.

\(\mathrm{A}\) ______ is a quantity that has both magnitude and direction.

If the rectangular coordinates of a point are (4,-6) the point symmetric to it with respect to the origin is ________.

Plot the point whose rectangular coordinates are (3,-1) . What quadrant does the point lie in?

If \(\mathbf{v}=a_{1} \mathbf{i}+b_{1} \mathbf{j}\) and \(\mathbf{w}=a_{2} \mathbf{i}+b_{2} \mathbf{j}\) are two vectors, then the _______ ______is defined as \(\mathbf{v} \cdot \mathbf{w}=a_{1} a_{2}+b_{1} b_{2}\).

If \(\mathbf{v}\) is a vector, then \(\mathbf{v}+(-\mathbf{v})=\) _____

\(\sin \frac{2 \pi}{3}=\)________; \(\cos \frac{4 \pi}{3}=\) ________.

A vector \(\mathbf{u}\) for which \(\|\mathbf{u}\|=1\) is called a(n) _____ vector.

If \(\mathbf{v}=3 \mathbf{w},\) then the two vectors \(\mathbf{v}\) and \(\mathbf{w}\) are _________.

Simplify: \(e^{2} \cdot e^{5}=\) ________ ;\(\left(e^{4}\right)^{3}=\) __________.

If \(\mathbf{v}=\langle a, b\rangle\) is an algebraic vector whose initial point is the origin, then \(\mathbf{v}\) is called a(n) _____ vector.

Is the sine function even, odd, or neither?

Draw the angle \(\frac{5 \pi}{6}\) in standard position.

True or False Given two nonzero, nonorthogonal vectors \(\mathbf{v}\) and \(\mathbf{w},\) it is always possible to decompose \(\mathbf{v}\) into two vectors, one parallel to \(\mathbf{w}\) and the other orthogonal to \(\mathbf{w}\).

In the complex plane, the \(x\) -axis is referred to as the _________ axis, and the \(y\) -axis is called the _______ axis.

\(\sin \frac{5 \pi}{4}=\) ______

If \(P=(a, b)\) is a point on the terminal side of the angle \(\theta\) at a distance \(r\) from the origin, then \(\tan \theta=\) ________.

If \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\) are two forces acting on an object, the vector sum \(\mathbf{F}_{1}+\mathbf{F}_{2}\) is called the _____ force.

\(\tan ^{-1}(-1)=\) _______.

Multiple Choice The angle \(\theta, 0 \leq \theta \leq \pi,\) between two nonzero vectors \(\mathbf{u}\) and \(\mathbf{v}\) can be found using what formula? (a) \(\sin \theta=\frac{\|\mathbf{u}\|}{\|\mathbf{v}\|}\) (b) \(\cos \theta=\frac{\|\mathbf{u}\|}{\|\mathbf{v}\|}\) (c) \(\sin \theta=\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}\) (d) \(\cos \theta=\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}\)

Suppose \(z_{1}=r_{1} e^{i \theta_{1}}\) and \(z_{2}=r_{2} e^{i \theta_{2}}\) are two complex numbers. Then \(z_{1} z_{2}=\) _______.

The origin in rectangular coordinates coincides with the ______ in polar coordinates; the positive \(x\) -axis in rectangular coordinates coincides with the _________ ___________ in polar coordinates.

Multiple Choice If two nonzero vectors \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal, then the angle between them has what measure? (a) \(\pi\) (b) \(\frac{\pi}{2}\) (c) \(\frac{3 \pi}{2}\) (d) \(2 \pi\)

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=\mathbf{i}-\mathbf{j}, \quad \mathbf{w}=\mathbf{i}+\mathbf{j} $$

Every nonzero complex number has exactly _______ distinct complex cube roots.

If \(\mathrm{v}\) is a vector with initial point \(\left(x_{1}, y_{1}\right)\) and terminal point \(\left(x_{2}, y_{2}\right),\) then which of the following is the position vector that equals \(\mathbf{v} ?\) (a) \(\left\langle x_{2}-x_{1}, y_{2}-y_{1}\right\rangle\) (b) \(\left\langle x_{1}-x_{2}, y_{1}-y_{2}\right\rangle\) (c) \(\left\langle\frac{x_{2}-x_{1}}{2}, \frac{y_{2}-y_{1}}{2}\right\rangle\) (d) \(\left\langle\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right\rangle\)

Multiple Choice In a rectangular coordinate system, where does the point with polar coordinates \(\left(1,-\frac{\pi}{2}\right)\) lie? (a) in quadrant IV (b) on the \(y\) -axis (c) in quadrant II (d) on the \(x\) -axis

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=\mathbf{i}+\mathbf{j}, \quad \mathbf{w}=-\mathbf{i}+\mathbf{j} $$

If \(\mathbf{v}\) is a nonzero vector with direction angle \(\alpha, 0^{\circ} \leq \alpha<360^{\circ},\) between \(\mathbf{v}\) and \(\mathbf{i},\) then \(\mathbf{v}\) equals which of the following? (a) \(\|\mathbf{v}\|(\cos \alpha \mathbf{i}-\sin \alpha \mathbf{j})\) (b) \(\|\mathbf{v}\|(\cos \alpha \mathbf{i}+\sin \alpha \mathbf{j})\) (c) \(\|\mathbf{v}\|(\sin \alpha \mathbf{i}-\cos \alpha \mathbf{j})\) (d) \(\|\mathbf{v}\|(\sin \alpha \mathbf{i}+\cos \alpha \mathbf{j})\)

Multiple Choice The point \(\left(5, \frac{\pi}{6}\right)\) can also be represented by which polar coordinates? (a) \(\left(5,-\frac{\pi}{6}\right)\) (b) \(\left(-5, \frac{13 \pi}{6}\right)\) (c) \(\left(5,-\frac{5 \pi}{6}\right)\) (d) \(\left(-5, \frac{7 \pi}{6}\right)\)

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{w}=\mathbf{i}-2 \mathbf{j} $$

Multiple Choice If \(z=x+y i\) is a complex number, then the magnitude of \(z\) is: (a) \(x^{2}+y^{2}\) (b) \(|x|+|y|\) (c) \(\sqrt{x^{2}+y^{2}}\) (d) \(\sqrt{|x|+|y|}\)

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=2 \mathbf{i}+2 \mathbf{j}, \quad \mathbf{w}=\mathbf{i}+2 \mathbf{j} $$

Multiple Choice If \(z_{1}=r_{1} e^{i \theta_{1}}\) and \(z_{2}=r_{2} e^{i \theta_{2}}\) are complex numbers, then \(\frac{z_{1}}{z_{2}}, z_{2} \neq 0,\) equals: (a) \(\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1}-\theta_{2}\right)}\) (b) \(\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1} \cdot \theta_{2}\right)}\) (c) \(\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1}+\theta_{2}\right)}\) (d) \(\frac{r_{1}}{r_{2}} e^{i\left(\theta_{1} / \theta_{2}\right)}\)

True or False A cardioid passes through the pole.

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=\sqrt{3} \mathbf{i}-\mathbf{j}, \quad \mathbf{w}=\mathbf{i}+\mathbf{j} $$

In Problems \(13-24,\) plot each complex number in the complex plane and write it in polar form and in exponential form. $$ 1+i $$

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=\mathbf{i}+\sqrt{3} \mathbf{j}, \quad \mathbf{w}=\mathbf{i}-\mathbf{j} $$

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ -1+i $$

In polar coordinates, the points \((r, \theta)\) and \((-r, \theta)\) are symmetric with respect to which of the following? (a) the polar axis (or \(x\) -axis) (b) the pole (or origin) (c) the line \(\theta=\frac{\pi}{2}\) (or \(y\) -axis) (d) the line \(\theta=\frac{\pi}{4}\) \((\) or \(y=x)\)

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=3 \mathbf{i}+4 \mathbf{j}, \quad \mathbf{w}=-6 \mathbf{i}-8 \mathbf{j} $$

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ \sqrt{3}-i $$

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r=4 $$

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=3 \mathbf{i}-4 \mathbf{j}, \quad \mathbf{w}=9 \mathbf{i}-12 \mathbf{j} $$

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ 1-\sqrt{3} i $$

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=4 \mathbf{i}, \quad \mathbf{w}=\mathbf{j} $$

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ -3 i $$

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=\mathbf{i}, \quad \mathbf{w}=-3 \mathbf{j} $$

Find \(a\) so that the vectors \(\mathbf{v}=\mathbf{i}-a \mathbf{j}\) and \(\mathbf{w}=2 \mathbf{i}+3 \mathbf{j}\) are orthogonal.

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ 4-4 i $$