Q: Suppose $$\mathbf{u} = 2\mathbf{i} + \mathbf{j} - 2\mathbf{k}$$ and $$\mathbf{v} = 3\mathbf{i} - 2\mathbf{j} - \mathbf{k}. $$What is the angle $$\theta$$ between $$\mathbf{u}$$ and $$\mathbf{v}$$?

$$\theta = \arccos\left(\dfrac{4}{\sqrt{9} \cdot \sqrt{14}}\right)$$

Q: If $$\mathbf{u}$$ and $$\mathbf{v}$$ are nonzero vectors in $$\mathbb{R}^3$$, which of the following statements is always true?

$$\mathbf{u} \cdot (\mathbf{u} \times \mathbf{v}) = 0$$

Question 1

Given vectors $$\mathbf{u}$$ and $$\mathbf{v} in$$ $$\mathbb{R}^3$$, with $$	heta$$ the angle between them, which formula correctly gives the scalar projection $$s of$$ $$\mathbf{u}$$ onto $$\mathbf{v}$$?

Accepted Answer

$$s = \dfrac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|}$$

Question 2

Suppose $$\mathbf{u} = 2\mathbf{i} + \mathbf{j} - 2\mathbf{k}$$ and $$\mathbf{v} = 3\mathbf{i} - 2\mathbf{j} - \mathbf{k}. $$What is the angle $$	heta$$ between $$\mathbf{u}$$ and $$\mathbf{v}$$?

Accepted Answer

$$	heta = \arccos\left(\dfrac{4}{\sqrt{9} \cdot \sqrt{14}}ight)$$

Question 3

Which of the following is the correct formula for the vector projection of $$\mathbf{u}$$ onto $$\mathbf{v}$$?

Accepted Answer

$$\mathrm{proj}_{\mathbf{v}} \mathbf{u} = \dfrac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2} \mathbf{v}$$

Question 4

Given the plane $$2x - 3y - 4z = 0$$, what is a normal vector to this plane?

Accepted Answer

$$\mathbf{n} = 2\mathbf{i} - 3\mathbf{j} - 4\mathbf{k}$$

Question 5

What is the distance from the point $$P_0 = (2$$, -1, 3)$$ to the plane 2x - 2y - z = 9$?

Accepted Answer

$$s = \dfrac{|4 + 2 - 3 - 9|}{\sqrt{4 + 4 + 1}} = 2$$

Question 6

Which equation represents the plane passing through the point $$(1,2,3)$$ and parallel to the plane $$3x + y - 2z = 15$$?

Accepted Answer

$$3x + y - 2z = 3 + 2 - 6 = -1$$

Question 7

Given three points $$A = (2, -1, -1)$$, $$B = (0, 1, -2)$$, and $$C = (1, -3, 1)$$, which method would you use to find the angle at $$A in $$triangle $$ABC$$?

Accepted Answer

Compute vectors $$\overrightarrow{AB}$$ and $$\overrightarrow{AC}$$, then use the dot product formula.

Question 8

Which of the following is the correct formula for the distance from a point $$P_0$$ = ($$x_0$$, $$y_0$$, $$z_0$$)$$ to the plane Ax + By + Cz = D$?

Accepted Answer

$$s = \dfrac{|Ax_0$ + By_0$ + Cz_0$ - D|}{\sqrt{A^2$ + B^2$ + C^2$}}$$

Question 9

If $$\mathbf{u}$$ and $$\mathbf{v}$$ are nonzero vectors in $$\mathbb{R}^3$$, which of the following statements is always true?

Accepted Answer

$$\mathbf{u} \cdot (\mathbf{u} 	imes \mathbf{v}) = 0$$

Question 10

Convert the spherical coordinates $$(R, \varphi, 	heta) = (4, \pi/3, 2\pi/3) to $$Cartesian coordinates $$(x, y, z). $$Which is correct?

Accepted Answer

$$(x, y, z) = (4 \sin(\pi/3) \cos(2\pi/3), 4 \sin(\pi/3) \sin(2\pi/3), 4 \cos(\pi/3))$$

Question 11

Which of the following equations describes a sphere in $$\mathbb{R}^3$$?

Accepted Answer

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

Question 12

Given $$\mathbf{u} = \mathbf{i} - \mathbf{j}$$ and $$\mathbf{v} = \mathbf{j} + 2\mathbf{k}$$, what is $$|\mathbf{u}|$$?

Accepted Answer

$$|\mathbf{u}| = \sqrt{2}$$

Question 13

Which of the following best describes the set of points satisfying $$x^2$$ + $$y^2$$ + $$z^2 = 2z$?

Accepted Answer

A sphere centered at $$(0, 0, 1)$$ with radius $$1$$

Question 14

Which coordinate system is most convenient for describing a sphere centered at the origin?

Accepted Answer

Spherical coordinates

Question 15

Given two planes $$x - y = 3$$ and $$x + y + z = 0$$, which vector is parallel to their line of intersection?

Accepted Answer

$$\mathbf{v} = (1, 1, -2)$$

Question 16

What is the geometric meaning of $$\mathbf{u} \cdot \mathbf{v} = 0$$?

Accepted Answer

$$\mathbf{u}$$ and $$\mathbf{v}$$ are perpendicular

Analytical Geometry in 3D: Vectors, Planes, and Projections

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