Q: If $$\textbf{a} = (2, 3)$$ and $$\textbf{b} = (-1, 4)$$, what is the angle $$\theta$$ between them? (Use $$\cos \theta = \frac{\textbf{a} \cdot \textbf{b}}{\|\textbf{a}\| \|\textbf{b}\|})$$

$$\theta = \cos$$^{-1}$$\left(\frac{10}{\sqrt{13} \sqrt{17}}\right)$$

Q: A complex number $$z = 3 + 4i is $$represented in polar form as $$r(\cos \theta + i \sin \theta). $$What are $$r$$ and $$\theta$$?

$$r = 5, \theta = \tan$$^{-1}$$\left(\frac{4}{3}\right)$$

Q: Which of the following is the correct formula for converting from polar to rectangular coordinates?

$$x = r \cos \theta, y = r \sin \theta$$

Q: If $$\textbf{a} = (1, 2)$$ and $$\textbf{b} = (2, -1)$$, are $$\textbf{a}$$ and $$\textbf{b}$$ orthogonal?

Yes, because $$\textbf{a} \cdot \textbf{b} = 0$$

Q: A force of $$15$$ pounds is applied at an angle of $$60^\circ to $$the direction of displacement of $$8$$ feet. How much work is done?

$$W = 15 \times 8 \times \cos 60^\circ = 60 ft-lb$$

Question 1

A vector $$	extbf{v}$$ has components $$(-3, 4). $$What is the magnitude of $$	extbf{v}$$?

Accepted Answer

$$\sqrt{(-3)^2$ + 4^2$} = 5$$

Question 2

Given vectors $$	extbf{a} = (2, -1)$$ and $$	extbf{b} = (1, 3)$$, what is $$	extbf{a} + 	extbf{b}$$?

Accepted Answer

$$(3, 2)$$

Question 3

A force of $$10$$ pounds is applied at an angle of $$30^\circ$$ above the horizontal. What is the horizontal component of the force?

Accepted Answer

$$10 \cos 30^\circ = 5 \sqrt{3}$$

Question 4

Which of the following is the correct formula for the dot product of $$	extbf{a} = (a_1$, a_2$)$$ and $$	extbf{b} = (b_1$, b_2$)$$?

Accepted Answer

$$a_1$$ $$b_1$$ + $$a_2$$ $$b_2$$

Question 5

If $$	extbf{a} = (2, 3)$$ and $$	extbf{b} = (-1, 4)$$, what is the angle $$	heta$$ between them? (Use $$\cos 	heta = \frac{	extbf{a} \cdot 	extbf{b}}{\|	extbf{a}\| \|	extbf{b}\|})$$

Accepted Answer

$$	heta = \cos$$^{-1}$$\left(\frac{10}{\sqrt{13} \sqrt{17}}ight)$$

Question 6

A complex number $$z = 3 + 4i is $$represented in polar form as $$r(\cos 	heta + i \sin 	heta). $$What are $$r$$ and $$	heta$$?

Accepted Answer

$$r = 5, 	heta = 	an$$^{-1}$$\left(\frac{4}{3}ight)$$

Question 7

Which statement is true about unit vectors?

Accepted Answer

A unit vector has a magnitude of $$1$$ and points in the same direction as the original vector.

Question 8

A boat heads due east at $$20$$ miles per hour while a current flows north at $$10$$ miles per hour. What is the magnitude of the resultant velocity?

Accepted Answer

$$\sqrt{20^2$ + 10^2$} = \sqrt{500} = 22.36$$ mph

Question 9

Given $$z_1 = 2$$(\cos 30^\circ + i \sin 30^\circ)$$ and z_2$ = 3(\cos 45^\circ + i \sin 45^\circ)$$, what is the product $$z_1$$ $$z_2 in $$polar form?

Accepted Answer

$$6(\cos 75^\circ + i \sin 75^\circ)$$

Question 10

Which of the following is the correct formula for converting from polar to rectangular coordinates?

Accepted Answer

$$x = r \cos 	heta, y = r \sin 	heta$$

Question 11

If $$	extbf{a} = (1, 2)$$ and $$	extbf{b} = (2, -1)$$, are $$	extbf{a}$$ and $$	extbf{b}$$ orthogonal?

Accepted Answer

Yes, because $$	extbf{a} \cdot 	extbf{b} = 0$$

Question 12

A force of $$15$$ pounds is applied at an angle of $$60^\circ to $$the direction of displacement of $$8$$ feet. How much work is done?

Accepted Answer

$$W = 15 	imes 8 	imes \cos 60^\circ = 60 ft-lb$$

Question 13

What is the rectangular form of the complex number $$z = 4 	ext{cis} 120^\circ$$?

Accepted Answer

$$z = 4(\cos 120^\circ + i \sin 120^\circ) = -2 + 2\sqrt{3}i$$

Question 14

Which of the following is De Moivre's Theorem for powers of complex numbers?

Accepted Answer

$$(r(\cos 	heta + i \sin 	heta))^n = r^n (\cos n	heta + i \sin n	heta)$$

Question 15

Find all cube roots of $$8 in $$polar form.

Accepted Answer

$$2(\cos 0^\circ + i \sin 0^\circ), 2(\cos 120^\circ + i \sin 120^\circ), 2(\cos 240^\circ + i \sin 240^\circ)$$

Question 16

A point in polar coordinates is given by $$(r, 	heta) = (5, 150^\circ). $$What are its rectangular coordinates?

Accepted Answer

$$(x, y) = (5 \cos 150^\circ, 5 \sin 150^\circ) = (-4.33, 2.5)$$

Question 17

Which of the following is the correct formula for the $$nth $$roots of a complex number $$z = r(\cos 	heta + i \sin 	heta)$$?

Accepted Answer

$$w_k = r$$^{1/n}$$(\cos \frac{	heta + 2\pi k}{n} + i \sin \frac{	heta + 2\pi k}{n})$$, $$k = 0, 1, ..., n-1$$

Question 18

A vector $$	extbf{v}$$ has magnitude $$10$$ and direction angle $$120^\circ. $$What are its horizontal and vertical components?

Accepted Answer

$$x = 10 \cos 120^\circ = -5, y = 10 \sin 120^\circ = 8.66$$

Vectors, the Complex Plane, and Polar Coordinates: Structured Study Notes

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