Question 1

Given the vector $$\vec{v} = \langle -4, -3 \rangle$$, what is the normalized (unit) vector $$\vec{u} in $$the same direction?

Accepted Answer

$$\langle -\frac{4}{5}, -\frac{3}{5} \rangle$$

Question 2

What is the magnitude of the vector $$\vec{a} = \langle 6, 8 \rangle$$?

Accepted Answer

$$10$$

Question 3

Which of the following is the equation of a sphere with center $$C = (2, -1, 5)$$ and radius $$r = 4$$?

Accepted Answer

$$(x-2)^2$$ + $$(y+1)^2$$ + $$(z-5)^2 = 16$

Question 4

If $$P = (0, 4, 3)$$ and $$Q = (-4, -2, 7)$$, what is the distance $$|\overrightarrow{PQ}|$$ between $$P$$ and $$Q$$?

Accepted Answer

$$\sqrt{(-4-0)^2$ + (-2-4)^2$ + (7-3)^2$} = \sqrt{16 + 36 + 16} = 10$$

Question 5

Which of the following describes the set of all points $$(x, y)$$ that are a distance $$r$$ from the center $$C = (a, b) in 2D$$?

Accepted Answer

$$(x-a)^2$$ + $$(y-b)^2$$ = $$r^2$$

Question 6

Given the vector $$\vec{w} = \langle 2, -5, 4 \rangle$$, what is its magnitude $$|\vec{w}|$$?

Accepted Answer

$$\sqrt{2^2$ + (-5)^2$ + 4^2$} = \sqrt{4 + 25 + 16} = \sqrt{45}$$

Question 7

A sphere has the equation $$x^2$$ + $$y^2$$ + $$z^2 + 8x - 6y + 9 = 11$. What is the center and radius of the sphere?

Accepted Answer

Center $$(-4, 3, 0)$$, radius $$6$$

Question 8

Which of the following is the correct formula for the distance between points $$P = (x_1$, y_1$, z_1$)$$ and $$Q = (x_2$, y_2$, z_2$) in 3D$$?

Accepted Answer

$$\sqrt{(x_2$-x_1$)^2$$ + ($$y_2$$-$$y_1)^2$ + (z_2$-z_1$)^2$$}$

Question 9

If $$\vec{v} = \langle 3, 4 \rangle$$, what is the unit vector in the direction of $$\vec{v}$$?

Accepted Answer

$$\langle \frac{3}{5}, \frac{4}{5} \rangle$$

Question 10

Which of the following is a unit vector in the $$z-$$direction in 3D space?

Accepted Answer

$$\vec{k} = \langle 0, 0, 1 \rangle$$

Question 11

A sphere in 3D has center $$C = (a, b, c)$$ and radius $$r. $$Which inequality describes the region inside the sphere?

Accepted Answer

$$(x-a)^2$$ + $$(y-b)^2$$ + $$(z-c)^2$$ \leq $$r^2$$

Question 12

If $$\vec{a} = \langle 2, 1, -2 \rangle$$ and $$\vec{b} = \langle -1, 3, 4 \rangle$$, what is $$\vec{a} + \vec{b}$$?

Accepted Answer

$$\langle 1, 4, 2 \rangle$$

Question 13

Which rule is used to determine the orientation of axes in a 3D coordinate system?

Accepted Answer

The right-hand rule

Question 14

If $$\vec{v} = \langle x, y, z \rangle$$, what is the formula for its magnitude?

Accepted Answer

$$|\vec{v}| = \sqrt{x^2$ + y^2$ + z^2$}$$

Question 15

Given two points $$P = (1, 2, 3)$$ and $$Q = (4, 6, 3)$$, what is the vector $$\overrightarrow{PQ}$$?

Accepted Answer

$$\langle 3, 4, 0 \rangle$$

Question 16

If $$\vec{v} = \langle 2, -1, 2 \rangle$$ and $$a = 3$$, what is $$a\vec{v}$$?

Accepted Answer

$$\langle 6, -3, 6 \rangle$$

Question 17

A sphere has diameter endpoints at $$P = (0, 4, 3)$$ and $$Q = (-4, -2, 7). $$What is the radius of the sphere?

Accepted Answer

$$r = \frac{1}{2}|\overrightarrow{PQ}| = 6$$

Question 18

Which of the following is the equation for a circle in 2D with center $$C = (a, b)$$ and radius $$r$$?

Accepted Answer

$$(x-a)^2$$ + $$(y-b)^2$$ = $$r^2$$

Question 19

If $$\vec{v} = \langle -2, 1, 2 \rangle$$, what is the direction vector from the origin to the point $$( -2, 1, 2 )$$?

Accepted Answer

$$\langle -2, 1, 2 \rangle$$

Vectors and Spheres in Calculus: Coordinate Geometry and Applications

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