Q: If $$\vec{a} = \langle 9, 7, 5 \rangle$$ and $$\vec{b} = \langle 8, 5, 1 \rangle$$, what is the dot product $$\vec{a} \cdot \vec{b}$$?

$$9 \times 8 + 7 \times 5 + 5 \times 1$$

Q: Convert the rectangular coordinates $$(6, 7) to $$polar coordinates.

$$(r, \theta) = (\sqrt{85}, \tan$$^{-1}$$(7/6))$$

Question 1

Given the vector $$\vec{a} = \langle 9, 7, 5 \rangle$$, what is the magnitude $$|\vec{a}|$$?

Accepted Answer

$$\sqrt{9^2$ + 7^2$ + 5^2$}$$

Question 2

Which of the following is the unit vector in the direction of $$\vec{c} = -3\vec{i} - 7\vec{j} + 4\vec{k}$$?

Accepted Answer

$$\frac{1}{\sqrt{74}}(-3\vec{i} - 7\vec{j} + 4\vec{k})$$

Question 3

If $$\vec{a} = \langle 9, 7, 5 \rangle$$ and $$\vec{b} = \langle 8, 5, 1 \rangle$$, what is the dot product $$\vec{a} \cdot \vec{b}$$?

Accepted Answer

$$9 	imes 8 + 7 	imes 5 + 5 	imes 1$$

Question 4

Given $$\vec{a} = \langle 9, 7, 5 angle$$ and $$\vec{b} = \langle 8, 5, 1 angle$$, what is the angle $$	heta$$ between $$\vec{a}$$ and $$\vec{b}$$?

Accepted Answer

$$\cos$$^{-1}$$\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|\,|\vec{b}|}\right)$$

Question 5

Convert the rectangular coordinates $$(6, 7) to $$polar coordinates.

Accepted Answer

$$(r, 	heta) = (\sqrt{85}, 	an$$^{-1}$$(7/6))$$

Question 6

Convert the polar coordinates $$(5, \frac{\pi}{3}) to $$rectangular coordinates.

Accepted Answer

$$(5\cos\frac{\pi}{3}, 5\sin\frac{\pi}{3})$$

Question 7

Express the complex number $$2 - 2\sqrt{3}i in $$polar form.

Accepted Answer

$$4\left(\cos\frac{5\pi}{3} + i\sin\frac{5\pi}{3}\right)$$

Question 8

Find all fourth roots of the complex number $$2 - 2\sqrt{3}i.$$

Accepted Answer

$$\sqrt{2}\left(\cos\frac{2\pi k + 5\pi}{12} + i\sin\frac{2\pi k + 5\pi}{12}\right)$$ for $$k = 0, 1, 2, 3$$

Question 9

Solve the equation $$x^3 = 27$$ \omega$$, where $$\omega$$ is a cube root of unity.

Accepted Answer

$$x = 3\omega$$

Question 10

Write $$\cos(3x) in $$terms of $$\cos(x)$$ and $$\sin(x).$$

Accepted Answer

$$\cos(3x) = 4\cos^3$(x) - 3\cos(x)$$

Question 11

A drone flies from the origin to the point $$(6, 7) in $$the $$xy-$$plane. What is the straight-line distance it travels?

Accepted Answer

$$\sqrt{6^2$ + 7^2$}$$

Question 12

Which of the following is the rectangular form of the polar point $$(r, 	heta) = (5, \frac{\pi}{4})$$?

Accepted Answer

$$(5\cos\frac{\pi}{4}, 5\sin\frac{\pi}{4})$$

Question 13

If $$\vec{a} = \langle 9, 7, 5 \rangle$$ and $$\vec{b} = \langle 8, 5, 1 \rangle$$, which of the following is the correct formula for the projection of $$\vec{a}$$ onto $$\vec{b}$$?

Accepted Answer

$$\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\vec{b}$$

Question 14

A medical imaging device uses the vector $$\vec{d} = 2\vec{i} - 2\sqrt{3}\vec{j}. $$What is the angle this vector makes with the positive $$x-$$axis?

Accepted Answer

$$	an$$^{-1}$$\left(\frac{-2\sqrt{3}}{2}ight) = -\frac{\pi}{3}$$

Question 15

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

Accepted Answer

$$64(\cos3	heta + i\sin3	heta)$$

Question 16

Which of the following is equivalent to $$\cos(x)\cos(y) - \sin(x)\sin(y)$$?

Accepted Answer

$$\cos(x + y)$$

Question 17

A satellite orbits the Earth in a path described by the equation $$x^2$$ + $$y^2 = 49$. What is the radius of its orbit?

Accepted Answer

$$7$$

Question 18

If $$z = r(\cos	heta + i\sin	heta)$$, which formula gives the $$nth $$roots of $$z$$?

Accepted Answer

$$\sqrt[n]{r}\left(\cos\frac{	heta + 2\pi k}{n} + i\sin\frac{	heta + 2\pi k}{n}ight)$$, $$k = 0, 1, ..., n-1$$

Vectors, Coordinate Systems, Complex Numbers, and Analytic Geometry: Precalculus Study Notes

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