Q: A plane contains the points $$A(1, 2, 3)$$, $$B(4, 2, 1)$$, and $$C(1, 5, 2). $$What is a normal vector to the plane?

$$\langle 3, 0, -2 \rangle \times \langle 0, 3, -1 \rangle$$

Question 1

Which of the following is the vector equation of a line passing through the point $$(2, -1, 3)$$ with direction vector $$\mathbf{m} = \langle 4, 5, 6 \rangle$$?

Accepted Answer

$$\mathbf{r} = \langle 2, -1, 3 \rangle + t\langle 4, 5, 6 \rangle$$

Question 2

Given the parametric equations $$x = 1 + 2t$$, $$y = 3 - t$$, $$z = 4 + 5t$$, what is the direction vector of the line?

Accepted Answer

$$\langle 2, -1, 5 \rangle$$

Question 3

Which of the following is the symmetric equation of the line with parametric equations $$x = 3 + t$$, $$y = 2 - 2t$$, $$z = 5 + 4t$$?

Accepted Answer

$$\dfrac{x - 3}{1} = \dfrac{y - 2}{-2} = \dfrac{z - 5}{4}$$

Question 4

A line has direction vector $$\mathbf{m} = \langle 2, -3, 1 \rangle. $$Which of the following vectors is perpendicular to $$\mathbf{m}$$?

Accepted Answer

$$\langle 3, 2, 0 \rangle$$

Question 5

Which of the following is the Cartesian equation of the line passing through $$(1, 2)$$ with slope $$m = 3$$?

Accepted Answer

$$y - 2 = 3(x - 1)$$

Question 6

Given two lines with direction vectors $$\mathbf{a} = \langle 1, 2, 3 \rangle$$ and $$\mathbf{b} = \langle 2, 4, 6 \rangle$$, what is the relationship between the lines?

Accepted Answer

The lines are parallel.

Question 7

Which of the following is the vector equation of a plane passing through the point $$(1, 2, 3)$$ with direction vectors $$\mathbf{a} = \langle 1, 0, 2 \rangle$$ and $$\mathbf{b} = \langle 0, 1, -1 \rangle$$?

Accepted Answer

$$\mathbf{r} = \langle 1, 2, 3 \rangle + s\langle 1, 0, 2 \rangle + t\langle 0, 1, -1 \rangle$$

Question 8

What is the Cartesian equation of the plane passing through $$(2, 1, -3)$$ with normal vector $$\mathbf{n} = \langle 1, -2, 4 \rangle$$?

Accepted Answer

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

Question 9

If two planes have normal vectors $$\mathbf{n}_1 = \langle 1, 2, 3 \rangle$$ and $$\mathbf{n}_2 = \langle 2, 4, 6 \rangle$$, what can be said about the planes?

Accepted Answer

The planes are parallel.

Question 10

A plane contains the points $$A(1, 2, 3)$$, $$B(4, 2, 1)$$, and $$C(1, 5, 2). $$What is a normal vector to the plane?

Accepted Answer

$$\langle 3, 0, -2 angle 	imes \langle 0, 3, -1 angle$$

Question 11

Which of the following is the parametric equation of a plane passing through $$(0, 0, 0)$$ with direction vectors $$\langle 1, 2, 0 \rangle$$ and $$\langle 0, 1, 3 \rangle$$?

Accepted Answer

$$\mathbf{r} = s\langle 1, 2, 0 \rangle + t\langle 0, 1, 3 \rangle$$

Question 12

Given the line $$\mathbf{r} = \langle 1, 2, 3 \rangle + t\langle 2, -1, 4 \rangle$$, what is the $$z-$$intercept of the line?

Accepted Answer

$$(0, 4, 7)$$

Question 13

Which of the following statements is true about the direction vector of a line?

Accepted Answer

Any nonzero scalar multiple of a direction vector is also a direction vector for the line.

Question 14

If a plane has the equation $$2x - 3y + z = 5$$, what is a normal vector to the plane?

Accepted Answer

$$\langle 2, -3, 1 \rangle$$

Question 15

Which of the following is a necessary condition for two lines in $$\mathbb{R}^3 to be $$perpendicular?

Accepted Answer

The dot product of their direction vectors is zero.

Equations of Lines and Planes: Vector, Parametric, and Cartesian Forms

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