What is the vector and parametric forms of the equation of the line passing through points/2022

Answer all questions in the space provided. Show ALL your steps in your calculations for

full marks. Answer in complete sentences when necessary.

1. What is the vector and parametric forms of the equation of the line passing through points

𝑃(−2,4) and 𝑄(4, −4)? [4K]

 

2. Four students each made a statement about the equation of a line in R2 given in vector form, 𝑟⃗ = (−3,5) +

𝑡(2, −3). Which students is/are correct? Justify your answer. [4A, 1C]

i. Eleven: The equation of this line in scalar form is 2𝑥 + 3𝑦 − 1 = 0

ii. Lucas: A line 𝐿, that is parallel to the given line is as follows:

𝐿:{

𝑥 = 1 + 2.5𝑡

𝑦 = 8 − 3.75𝑡

iii. Finn: A vector that is perpendicular to the given line is 𝑣⃗ = (9, −4)

iv. Dustin: Two points on this line are: (−1,2) and (0, 0.5)

K T C A

11 9 2 8

35% 15% 15% 35%

 

3. A point 𝑃(3,0,5) lies on a plane that can be defined by the scalar equation 𝐴𝑥 + 𝐵𝑦 + 𝐶𝑧 + 𝐷 = 0. A normal

vector, 𝑛⃗⃗, to this plane is (2, −3,1). Find the scalar equation. [3K]

4. How many solutions are there between the plane 𝑃1: (𝑥, 𝑦, 𝑧) = (4, −3, −1) + 𝑠(1, −3,1) + 𝑡(2,4, −3) and the

line 𝐿1: (𝑥, 𝑦, 𝑧, ) = (3,1 − 2) + 𝑘(−1, −1,1)? If there is/are solutions, find the value of the solution(s). [4K]

5. How many solutions are there between planes 𝑃1: 2𝑥 − 𝑦 + 2𝑧 = 2 and 𝑃2 : − 𝑥 + 2𝑦 + 𝑧 = 1? If there is/are

solutions, find the value of the solution(s). [4A]

 

6. Dustin is trying to design a weapon to fight the Demogorgon. His model has three planes, and he needs to figure

out their configuration. He thinks they intersect in a line or at a point. Is he correct? Justify your answer. If there

is/are solutions, find the value of the solution(s) [5T]

𝑃1: 𝑥 + 4𝑦 + 3𝑧 − 5 = 0

𝑃2: 𝑥 + 3𝑦 + 2𝑧 − 4 = 0

𝑃3: 𝑥 + 𝑦 − 𝑧 + 1 = 0

 

7. Lucas does not like Dustin weapon and decides to design his own. His model has a plane that contains the point

𝐴(−5,2, −4) and two direction vectors, 𝐴𝐵⃗⃗⃗⃗⃗⃗ = (4,8,1)and 𝐴𝐶⃗⃗⃗⃗⃗⃗ = (−3, −2,0). Which of the following statement(s)

about this plane is/are true? Justify your answer. [4T,1C]

i. The plane has a parametric equation that can be defined as follows:

{

𝑥 = −5 + 4𝑡 − 3𝑠

𝑦 = 2 + 8𝑡 − 2𝑠

𝑧 = −4 + 𝑡 + 𝑠𝑠

ii. The plane has a vector equation that can be defined as (𝑥, 𝑦, 𝑧) = (−5,2, −4) + 𝑡(3,2,0) +

𝑠(4,8,1)

iii. The point 𝐷(31,6,0) lies on the plane

iv. The 𝑧-intercept of the plane is 0,0, −5)