Question 1

Given the vector from $$P(3, 2) to$$ $$Q(-2, 1)$$, what is the position vector $$\overrightarrow{PQ}$$?

Accepted Answer

$$\langle -5, -1 \rangle$$

Question 2

What is the magnitude of the vector $$\overrightarrow{PQ}$$ from $$P(3, 2) to$$ $$Q(-2, 1)$$?

Accepted Answer

$$\sqrt{26}$$

Question 3

Which of the following is a unit vector in the direction of $$\overrightarrow{PQ}$$ from $$P(3, 2) to$$ $$Q(-2, 1)$$?

Accepted Answer

$$\left\langle \frac{-5}{\sqrt{26}}, \frac{-1}{\sqrt{26}} \right\rangle$$

Question 4

Let $$u = \langle 4, -2 \rangle$$, $$v = \langle -4, 6 \rangle$$, and $$w = \langle 0, 8 \rangle. $$What is $$10u - 3v + w$$?

Accepted Answer

$$\langle 52, -24 \rangle$$

Question 5

An airplane is flying due north at 560 mph. There is a wind blowing from the east at 75 mph. What is the airplane’s ground speed?

Accepted Answer

$$\sqrt{560^2$ + 75^2$}$$ mph

Question 6

What is the equation of a sphere centered at $$(2, -1, 3)$$ with radius $$5$$?

Accepted Answer

$$(x-2)^2$$ + $$(y+1)^2$$ + $$(z-3)^2 = 25$

Question 7

Let $$u = \langle 1, 1, -6 \rangle$$, $$v = \langle 3, 0, 8 \rangle$$, and $$w = \langle -3, -1, 2 \rangle. $$What is $$3u + 2(v-w)$$?

Accepted Answer

$$\langle 13, 5, -2 \rangle$$

Question 8

Given the forces $$F_1 = 20$$\hat{i} - 10\hat{j}$$, F_2$ = 30\hat{j} + 10\hat{k}$$, and $$F_3 = 40$$\hat{i} + 20\hat{k}$$ acting on an object at the origin, what is the magnitude of the combined force?

Accepted Answer

$$\sqrt{60^2$ + 20^2$ + 30^2$}$$

Question 9

Which of the following represents a ball centered at $$(1, -2, 3)$$ with radius $$4$$?

Accepted Answer

$$(x-1)^2$$ + $$(y+2)^2$$ + $$(z-3)^2$$ \leq 16$$

Question 10

If $$u = \langle 2, -3, 4 \rangle$$ and $$v = \langle -1, 5, 2 \rangle$$, what is $$u + v$$?

Accepted Answer

$$\langle 1, 2, 6 \rangle$$

Question 11

Which of the following statements is TRUE about the zero vector $$\vec{0}$$?

Accepted Answer

The zero vector is parallel to all vectors.

Question 12

Suppose $$u = \langle 2, 3 \rangle$$ and $$v = \langle -4, 1 \rangle. $$Which vector is parallel to $$u$$?

Accepted Answer

$$\langle 4, 6 \rangle$$

Question 13

Given $$u = \langle 1, 2, 3 \rangle$$ and $$v = \langle 4, -1, 2 \rangle$$, what is $$u - v$$?

Accepted Answer

$$\langle -3, 3, 1 \rangle$$

Question 14

A circle in the $$xy-$$plane is given by $$(x-2)^2$$ + $$(y+3)^2 = 9$. What does (x-2)^2$ + (y+3)^2$ \leq 9$$ represent?

Accepted Answer

The interior and boundary of the circle centered at $$(2, -3)$$ with radius $$3.$$

Question 15

If $$u = \langle 2, -1, 2 \rangle$$, what is the magnitude $$|u|$$?

Accepted Answer

$$\sqrt{9}$$

Vectors and Geometry in 3-Space: Study Notes

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