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) +

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)

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)