Q: Given two planes with normal vectors $$\vec{n}_1$$ and $$\vec{n}_2$$, the angle $$\theta$$ between the planes is given by which formula?

$$\cos \theta = \frac{\vec{n}_1 \cdot \vec{n}_2}{|\vec{n}_1||\vec{n}_2|}$$

Question 1

Which of the following is the correct vector equation for a line in 3D passing through the point $$(x_0$, y_0$, z_0$)$$ and parallel to the direction vector $$\vec{d} = \langle a, b, c \rangle$$?

Accepted Answer

$$\vec{r} = \langle x_0$, y_0$, z_0$ \rangle + t \langle a, b, c \rangle$$

Question 2

Given the parametric equations $$x = 2 + 3t$$, $$y = -4t$$, $$z = 5 + t$$, which of the following points lies on the line?

Accepted Answer

$$(5, -6, 6)$$

Question 3

Which of the following is the symmetric equation for a line passing through $$(x_0$, y_0$, z_0$)$$ and parallel to $$\langle a, b, c \rangle$$?

Accepted Answer

$$\displaystyle \frac{x - x_0$}{a} = \frac{y - y_0$}{b} = \frac{z - z_0$}{c}$$

Question 4

A plane passes through the point $$(2, 4, 1)$$ and has normal vector $$\vec{n} = \langle 2, 3, 4 \rangle. $$Which of the following is its scalar equation?

Accepted Answer

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

Question 5

Which of the following statements is TRUE about two lines in 3D space?

Accepted Answer

Two lines are skew if they do not intersect and are not parallel.

Question 6

Find the point of intersection between the line $$x = 2 + 3t$$, $$y = -4t$$, $$z = 5 + t$$ and the plane $$4x + 5y - 2z = 18.$$

Accepted Answer

$$(5, -6, 6)$$

Question 7

Given two planes with normal vectors $$\vec{n}_1$$ and $$\vec{n}_2$$, the angle $$	heta$$ between the planes is given by which formula?

Accepted Answer

$$\cos 	heta = \frac{\vec{n}_1 \cdot \vec{n}_2}{|\vec{n}_1||\vec{n}_2|}$$

Question 8

Which of the following best describes the geometric meaning of the normal vector $$\vec{n} to a $$plane?

Accepted Answer

$$\vec{n} is $$perpendicular to every vector lying in the plane.

Question 9

What is the distance between the parallel planes $$10x + 2y - 2z = 5$$ and $$5x + y - z = 1$$?

Accepted Answer

$$\displaystyle D = \frac{|5 - 1|}{\sqrt{5^2$ + 1^2$ + (-1)^2$}}$$

Question 10

A line passes through $$(1, 2, 3)$$ and is parallel to the vector $$\langle 4, -1, 2 \rangle. $$Which of the following is a correct set of parametric equations for the line?

Accepted Answer

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

Question 11

Which of the following is the scalar equation of a plane with normal vector $$\langle a, b, c \rangle$$ passing through $$(x_0$, y_0$, z_0$)$$?

Accepted Answer

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

Question 12

If two planes are not parallel, what is true about their intersection?

Accepted Answer

They intersect in a straight line.

Question 13

Given the plane $$2x + 3y + 4z = 9$$, what is the $$z-$$intercept of the plane?

Accepted Answer

$$(0, 0, 2.25)$$

Question 14

Which of the following is a necessary condition for two planes to be parallel?

Accepted Answer

Their normal vectors are parallel.

Question 15

A line in 3D is given by $$\vec{r} = \vec{r}_0 + t\vec{d}. $$Which of the following best describes $$\vec{d}$$?

Accepted Answer

$$\vec{d} is $$the direction vector of the line.

Question 16

Suppose a plane has equation $$x + y + z = 1. $$Which of the following is a normal vector to this plane?

Accepted Answer

$$\langle 1, 1, 1 \rangle$$

Question 17

Which of the following is the correct formula for the distance $$D$$ between the parallel planes $$ax + by + cz = d_1$ and ax + by + cz$$ = $$d_2$$?

Accepted Answer

$$D = \frac{|d_1$ - d_2$|}{\sqrt{a^2$ + b^2$ + c^2$}}$$

Question 18

A real-world application: An engineer needs to find the shortest distance from a point $$(3, 2, 1) to $$the plane $$2x - y + 2z = 5. $$Which formula should be used?

Accepted Answer

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

Section 2.5 - Equations of Lines and Planes

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