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 the line passing through $$P_0(2$$,3,4)$$ in the direction of $$\vec{v} = \langle -1, -2, -3 \rangle$$?

Accepted Answer

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

Question 6

Given the line $$\vec{r} = (1, -1, 0) + t(1, 2, -3)$$, 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 standard form of the equation of a plane with normal vector $$\vec{n} = (a, b, c)$$ passing through point $$(x_0$, y_0$, z_0$)$$?

Accepted Answer

$$a(x-x_0$) + b(y-y_0$) + c(z-z_0$) = 0$$

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

Which of the following statements is true about two planes with normal vectors $$\vec{n}_1$$ and $$\vec{n}_2$$?

Accepted Answer

The planes are parallel if $$\vec{n}_1$$ and $$\vec{n}_2$$ are parallel.

Question 11

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

Accepted Answer

The line $$(2, -1, 0) + t(1, -2, 1)$$

Question 12

If $$\vec{u}$$ and $$\vec{v}$$ are nonzero vectors and $$\vec{u} \cdot \vec{v} = 0$$, what can be said about the angle between them?

Accepted Answer

The vectors are orthogonal.

Question 13

Which of the following is a parametric equation for the line passing through $$(1, -2, 2) in $$the direction $$(1, 1, 1/2)$$?

Accepted Answer

$$x = 1 + t, y = -2 + t, z = 2 + t/2$$

Question 14

Suppose a plane has equation $$2x - 3y + z = 7. $$What is a normal vector to this plane?

Accepted Answer

$$(2, -3, 1)$$

Question 15

If $$\vec{u} = \langle 0, 0, 0 \rangle$$ and $$\vec{v} is $$any vector, what is $$\vec{u} \cdot \vec{v}$$?

Accepted Answer

$$0$$

Vectors, Lines, and Planes in Three-Dimensional Space

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