Question 1

Which of the following equations represents the sphere centered at $(2, -1, 3)$ with radius $5$?

Accepted Answer

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

Question 2

Given points $A(1,2,3)$ and $B(4,6,8)$, what is the component form of the vector $\overrightarrow{AB}$?

Accepted Answer

$(3, 4, 5)$

Question 3

A force $\vec{F}$ of magnitude $20$ lb acts at an angle of $45^\circ$ to the horizontal. What is the effective horizontal component of the force?

Accepted Answer

$20 \cos 45^\circ$

Question 4

Which of the following is a unit vector in the direction of $\vec{v} = (6, 2, -3)$?

Accepted Answer

$\left(\frac{6}{7}, \frac{2}{7}, \frac{-3}{7}\right)$

Question 5

Find the dot product of $\vec{u} = (2, -1, 4)$ and $\vec{v} = (0, 3, -2)$.

Accepted Answer

$-8$

Question 6

If $\vec{u} = (1, 2, 3)$ and $\vec{v} = (4, 5, 6)$, what is the angle $	heta$ between $\vec{u}$ and $\vec{v}$? (Use $\cos 	heta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}||\vec{v}|}$)

Accepted Answer

$\cos^{-1}\left(\frac{32}{\sqrt{14}\sqrt{77}}\right)$

Question 7

Which of the following statements is true about the cross product $\vec{u} 	imes \vec{v}$?

Accepted Answer

It is a vector perpendicular to both $\vec{u}$ and $\vec{v}$.

Question 8

Given $\vec{u} = (1, 0, 0)$ and $\vec{v} = (0, 1, 0)$, what is $\vec{u} 	imes \vec{v}$?

Accepted Answer

$(0, 0, 1)$

Question 9

The area of the parallelogram determined by vectors $\vec{u}$ and $\vec{v}$ is given by:

Accepted Answer

$|\vec{u} 	imes \vec{v}|$

Question 10

Find the equation of the plane passing through $(1, 2, 3)$ and perpendicular to the vector $\vec{n} = (4, -2, 5)$.

Accepted Answer

$4(x-1) - 2(y-2) + 5(z-3) = 0$

Question 11

What is the distance from the point $S(2, -1, 5)$ to the plane $3x + 2y + 6z = 6$?

Accepted Answer

$\frac{|3(2) + 2(-1) + 6(5) - 6|}{\sqrt{3^2 + 2^2 + 6^2}}$

Question 12

Which of the following is the parametric equation for the line passing through $(0, 1, -2)$ and parallel to $\vec{v} = (3, -1, 4)$?

Accepted Answer

$x = 0 + 3t, y = 1 - t, z = -2 + 4t$

Question 13

A jet airliner flying due east at $500$ mph encounters a $70$ mph tailwind blowing $60^\circ$ north of east. What is the new ground speed of the airplane?

Accepted Answer

$\sqrt{(500 + 70 \cos 60^\circ)^2 + (70 \sin 60^\circ)^2}$

Question 14

Which of the following is the equation of an ellipsoid centered at $(1, -2, 0)$ with axes lengths $2$, $2$, and $1$?

Accepted Answer

$\frac{(x-1)^2}{4} + \frac{(y+2)^2}{4} + z^2 = 1$

Question 15

The equation $y = x^2$ describes which type of surface in three-dimensional space?

Accepted Answer

A cylinder whose axis is parallel to the $z$-axis

Question 16

If $\vec{u} = (1, -2, 1)$ and $\vec{v} = (2, 4, -2)$, which statement is true?

Accepted Answer

$\vec{u}$ and $\vec{v}$ are parallel

Question 17

The volume of the parallelepiped determined by vectors $\vec{u}$, $\vec{v}$, and $\vec{w}$ is given by:

Accepted Answer

$|\vec{u} \cdot (\vec{v} 	imes \vec{w})|$

Vectors and the Geometry of Space – Study Notes (Thomas’ Calculus, Ch. 12.1)

Study Guide - Practice Questions

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