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)

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