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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