Question 1

Which of the following is the correct point-parallel form equation for a line in $$\mathbb{R}^2$$ passing through point $$P = (1,2)$$ with direction vector $$\vec{v} = (3,1)$$?

Accepted Answer

$$\vec{r}(t) = (1,2) + t(3,1)$$

Question 2

Given two points $$P = (2, -1)$$ and $$Q = (5, 3) in$$ $$\mathbb{R}^2$$, what is the two-point form equation for the line passing through both points?

Accepted Answer

$$\vec{r}(t) = (1-t)(2,-1) + t(5,3)$$

Question 3

If a line in $$\mathbb{R}^2$$ has parametric equations $$x = 4 + 2t$$, $$y = -3 + 5t$$, what is a direction vector for this line?

Accepted Answer

$$(2,5)$$

Question 4

Which of the following statements about direction vectors is TRUE?

Accepted Answer

Any nonzero vector parallel to a line is a direction vector for that line.

Question 5

Given the point-normal form equation for a line in $$\mathbb{R}^2$$: $$\vec{n} \cdot (\vec{r} - \vec{p}) = 0$$, what does $$\vec{n}$$ represent?

Accepted Answer

A normal vector perpendicular to the line

Question 6

Suppose a line in $$\mathbb{R}^2$$ passes through $$P = (0,1)$$ and is parallel to the vector $$\vec{v} = (2,-1). $$What is the parametric equation for this line?

Accepted Answer

$$\vec{r}(t) = (0,1) + t(2,-1)$$

Question 7

If the parametric equations of a line are $$x = 1 + 3t$$, $$y = 2 - 4t$$, which of the following is a point on the line when $$t = 2$$?

Accepted Answer

$$(7, -6)$$

Question 8

Which of the following is the correct point-normal form equation for a line passing through $$P = (1,2)$$ with normal vector $$\vec{n} = (-1,1)$$?

Accepted Answer

$$(-1,1) \cdot ((x,y) - (1,2)) = 0$$

Question 9

A line in $$\mathbb{R}^2 is $$given by the parametric equations $$x = 2 + t$$, $$y = 3 - 2t. $$Which of the following is a direction vector for this line?

Accepted Answer

$$(1, -2)$$

Question 10

Given two points $$P = (1,2)$$ and $$Q = (4,8)$$, what is the direction vector from $$P to$$ $$Q$$?

Accepted Answer

$$(3,6)$$

Question 11

If a line in $$\mathbb{R}^2$$ has point-normal form $$(-2,1) \cdot ((x,y) - (3,-4)) = 0$$, which of the following is a normal vector to the line?

Accepted Answer

$$(-2,1)$$

Question 12

Which of the following is TRUE about parametric equations of lines in $$\mathbb{R}^2$$?

Accepted Answer

They always contain a parameter $$t$$ and a direction vector.

Question 13

Suppose you are given the parametric equations $$x = 5 - 2t$$, $$y = 7 + 3t. $$What is the point on the line when $$t = -1$$?

Accepted Answer

$$(7, 4)$$

Question 14

Which of the following is the correct two-point form equation for the line passing through $$P = (0,0)$$ and $$Q = (2,3)$$?

Accepted Answer

$$\vec{r}(t) = (1-t)(0,0) + t(2,3)$$

Question 15

If a line in $$\mathbb{R}^2$$ passes through $$P = (2,1)$$ and is perpendicular to the vector $$\vec{n} = (4,-3)$$, which of the following is the correct point-normal form equation?

Accepted Answer

$$(4,-3) \cdot ((x,y) - (2,1)) = 0$$

Question 16

Which of the following is a valid parametric equation for a line passing through $$(1,2)$$ and parallel to $$(5,-2)$$?

Accepted Answer

$$(x, y) = (1,2) + t(5,-2)$$

Question 17

Given the parametric equations $$x = 3 + 4t$$, $$y = 2 - t$$, what is the direction vector and a point on the line?

Accepted Answer

Direction vector: $$(4, -1)$$; Point: $$(3, 2)$$

Question 18

If a line in $$\mathbb{R}^2 is $$given by the point-normal form $$(-1,2) \cdot ((x,y) - (3,4)) = 0$$, which of the following points lies on the line?

Accepted Answer

$$(5,3)$$

Lines and Planes: Equations and Forms (Math 1229A/B Unit 3 Study Notes)

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