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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

π(β2,4) and π(4, β4)? [4K]

2. Four students each made a statement about the equation of a line in R2 given in vector form, πβ = (β3,5) +

i. Eleven: The equation of this line in scalar form is 2π₯ + 3π¦ β 1 = 0

ii. Lucas: A line πΏ, that is parallel to the given line is as follows:

πΏ:{

π₯ = 1 + 2.5π‘

π¦ = 8 β 3.75π‘

iii. Finn: A vector that is perpendicular to the given line is π£β = (9, β4)

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

K T C A

11 9 2 8

35% 15% 15% 35%

3. A point π(3,0,5) lies on a plane that can be defined by the scalar equation π΄π₯ + π΅π¦ + πΆπ§ + π· = 0. A normal

vector, πββ, to this plane is (2, β3,1). Find the scalar equation. [3K]

4. How many solutions are there between the plane π1: (π₯, π¦, π§) = (4, β3, β1) + π (1, β3,1) + π‘(2,4, β3) and the

line πΏ1: (π₯, π¦, π§, ) = (3,1 β 2) + π(β1, β1,1)? If there is/are solutions, find the value of the solution(s). [4K]

5. How many solutions are there between planes π1: 2π₯ β π¦ + 2π§ = 2 and π2 : β π₯ + 2π¦ + π§ = 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]

π1: π₯ + 4π¦ + 3π§ β 5 = 0

π2: π₯ + 3π¦ + 2π§ β 4 = 0

π3: π₯ + π¦ β π§ + 1 = 0

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

π΄(β5,2, β4) and two direction vectors, π΄π΅ββββββ = (4,8,1)and π΄πΆββββββ = (β3, β2,0). Which of the following statement(s)

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

{

π₯ = β5 + 4π‘ β 3π

π¦ = 2 + 8π‘ β 2π

π§ = β4 + π‘ + π π

ii. The plane has a vector equation that can be defined as (π₯, π¦, π§) = (β5,2, β4) + π‘(3,2,0) +

π (4,8,1)

iii. The point π·(31,6,0) lies on the plane

iv. The π§-intercept of the plane is 0,0, β5)