Q: If a line is defined by $$x = x_0$ + a t$$, $$y = y_0$ + b t$$, $$z = z_0$ + c t$$, what condition must be satisfied for the standard form $$\frac{x - x_0$}{a} = \frac{y - y_0$}{b} = \frac{z - z_0$}{c} to be $$valid?

$$a \neq 0, b \neq 0, c \neq 0$$

Question 1

Which of the following is the correct vector equation for a plane given a point $$P_0$$ = ($$x_0$$, $$y_0$$, $$z_0$$)$$ and a normal vector $$\vec{n} = A\hat{i} + B\hat{j} + C\hat{k}$$?

Accepted Answer

$$A(x - x_0$) + B(y - y_0$) + C(z - z_0$) = 0$$

Question 2

Given three points $$P_1 = (1$$, 1, 0)$$, P_2$ = (2, 0, 2)$$, and $$P_3 = (0$$, 3, 3)$$, what is the normal vector $$\vec{n}$$ to the plane containing these points?

Accepted Answer

$$\vec{n} = (-7, -5, 1)$$

Question 3

What is the equation of the plane passing through $$(1, 0, 5)$$ with normal vector $$\vec{n} = 3\hat{i} + \hat{j} - \hat{k}$$?

Accepted Answer

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

Question 4

Which of the following best describes the relationship between the vector $$\vec{P_0P}$$ and the normal vector $$\vec{n}$$ for a point $$P on a $$plane?

Accepted Answer

$$\vec{P_0P} is $$orthogonal to $$\vec{n}$$

Question 5

Given the plane equation $$Ax + By + Cz = D$$, what does $$D$$ represent?

Accepted Answer

$$D = Ax_0$ + By_0$ + Cz_0$ for a point $$($$x_0$$, $$y_0$$, $$z_0$$)$$ on the plane

Question 6

Which of the following is the correct vector equation for a line given a point $$P_0$$ = ($$x_0$$, $$y_0$$, $$z_0$$)$$ and a direction vector $$\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}$$?

Accepted Answer

$$\vec{r} = \vec{r}_0 + t \cdot \vec{v}$$

Question 7

What is the standard form of the line passing through $$(3, 0, 3)$$ with direction vector $$(3, -2, 1)$$?

Accepted Answer

$$\frac{x - 3}{3} = \frac{y}{-2} = \frac{z - 3}{1}$$

Question 8

If a line is defined by $$x = x_0$ + a t$$, $$y = y_0$ + b t$$, $$z = z_0$ + c t$$, what condition must be satisfied for the standard form $$\frac{x - x_0$}{a} = \frac{y - y_0$}{b} = \frac{z - z_0$}{c} to be $$valid?

Accepted Answer

$$a 
eq 0, b 
eq 0, c 
eq 0$$

Question 9

Given two planes $$x + y - z = 0$$ and $$y + 2z = 6$$, what is the direction vector of their intersection line?

Accepted Answer

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

Question 10

A plane is defined by the equation $$2x - 3y + 4z = 12. $$Which of the following points lies on the plane?

Accepted Answer

$$(5, 2, 1)$$

Question 11

If a line is defined by $$\vec{r} = \vec{r}_0 + t \cdot \vec{v}$$, which of the following statements is true?

Accepted Answer

The line passes through $$\vec{r}_0$$ and is parallel to $$\vec{v}$$

Question 12

Which of the following is a valid method to find the normal vector to a plane given three points $$P_1$$, $$P_2$$, and $$P_3$$?

Accepted Answer

Take the cross product of $$\vec{P_1P_2}$$ and $$\vec{P_1P_3}$$

Question 13

A line in space is given by $$x = 1 + 2t$$, $$y = 3 - t$$, $$z = 4 + 5t. $$Which of the following points lies on the line?

Accepted Answer

$$(3, 2, 9)$$

Question 14

Which of the following best describes the intersection of two non-parallel planes?

Accepted Answer

A line

Question 15

If the direction vector of a line is $$(0, 2, 3)$$ and the point on the line is $$(1, 4, 5)$$, what is the parametric equation for the line?

Accepted Answer

$$x = 1$$, $$y = 4 + 2t$$, $$z = 5 + 3t$$

Question 16

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

Accepted Answer

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

Question 17

Which of the following is NOT a valid representation of a line in space?

Accepted Answer

$$x = x_0$ + a t$$, $$y = y_0$ + b t$$, $$z = z_0$ - c t$$

Question 18

If a plane has normal vector $$(A, B, C)$$ and passes through $$(x_0$, y_0$, z_0$)$$, which of the following points is guaranteed to NOT lie on the plane?

Accepted Answer

$$(x_0$ + 2A, y_0$ + 2B, z_0$ + 2C)$$

Planes and Lines in Three-Dimensional Space

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