Q: Given vectors $$\vec{u} = \langle 2, -1, 3 \rangle$$ and $$\vec{v} = \langle 1, 4, -2 \rangle$$, what is their dot product?

$$2 \times 1 + (-1) \times 4 + 3 \times (-2) = 2 - 4 - 6 = -8$$

Question 1

Given vectors $$\vec{u} = \langle 2, -1, 3 \rangle$$ and $$\vec{v} = \langle 1, 4, -2 \rangle$$, what is their dot product?

Accepted Answer

$$2 	imes 1 + (-1) 	imes 4 + 3 	imes (-2) = 2 - 4 - 6 = -8$$

Question 2

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

Accepted Answer

$$\mathrm{proj}_{\vec{v}} \vec{u} = \dfrac{\vec{u} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} \vec{v}$$

Question 3

If $$\vec{u} = \langle 1, 2, 3 angle$$ and $$\vec{v} = \langle 4, 5, 6 angle$$, what is $$\vec{u} 	imes \vec{v}$$?

Accepted Answer

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

Question 4

What is the area of the triangle with vertices $$P(1,0,1)$$, $$Q(2,2,4)$$, and $$R(2,1,2)$$?

Accepted Answer

$$\dfrac{\sqrt{6}}{2}$$

Question 5

Which of the following is the vector equation of a line passing through $$P_0$$($$x_0$$, $$y_0$$, $$z_0$$)$$ in the direction of $$\vec{v} = \langle a, b, c \rangle$$?

Accepted Answer

$$(x, y, z) = (x_0$, y_0$, z_0$) + t \langle a, b, c \rangle$$

Question 6

Given the line $$\vec{r} = \langle 1, -1, 0 \rangle + t \langle 1, 2, -3 \rangle$$, what is the distance from the point $$Q = (3, 3, -6) to $$the line?

Accepted Answer

$$0$$

Question 7

Which of the following is the general form of the equation of a plane in $$\mathbb{R}^3$$?

Accepted Answer

$$ax + by + cz = d$$

Question 8

A plane $$ax + by + cz = d$$ passes through the origin if and only if:

Accepted Answer

$$d = 0$$

Question 9

Find the equation of the plane passing through the points $$P(1, -2, 1)$$, $$Q(-1, 0, 1)$$, and the origin.

Accepted Answer

$$x + y + z = 0$$

Question 10

Two planes are orthogonal if:

Accepted Answer

The dot product of their normal vectors is zero.

Question 11

Given the planes $$x + 2y + 3z = 0$$ and $$x + y + z = 1$$, what is their intersection?

Accepted Answer

The line $$\langle 2, -1, 0 \rangle + t \langle 1, -2, 1 \rangle$$

Question 12

If $$\vec{n}_1$$ and $$\vec{n}_2$$ are normal vectors of two planes, which condition guarantees the planes are parallel?

Accepted Answer

$$\vec{n}_1 is a $$scalar multiple of $$\vec{n}_2$$

Question 13

Which of the following best describes the geometric meaning of the cross product $$\vec{u} 	imes \vec{v}$$?

Accepted Answer

A vector orthogonal to both $$\vec{u}$$ and $$\vec{v}$$ whose magnitude equals the area of the parallelogram formed by $$\vec{u}$$ and $$\vec{v}.$$

Question 14

Given the line $$x - 1 = y + 2 = 2z - 4$$, what is its vector form?

Accepted Answer

$$\langle 1, -2, 2 \rangle + t \langle 1, 1, \frac{1}{2} \rangle$$

Question 15

Suppose you are given a plane with normal vector $$\vec{n} = \langle 1, 1, 1 \rangle$$ passing through $$Q = (5, -3, 7). $$What is the equation of the plane?

Accepted Answer

$$x + y + z = 9$$

Vectors, Lines, and Planes in Three-Dimensional Space

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