Question 1 (27 marks) A line passes through the pounts P(. 2, -1) and Q(. -3,4)

Question

Question 1 (27 marks) A line passes through the pounts P(. 2, -1) and Q(. -3,4) Another line passes through the (a) Fund the vector equation for the lines PQ and MN points M(l. 4. 6) and NB, 5, 7 (8 marks) (b) Do lines PQ and MN intersect? If your answer is "yes", find the intersection point If your answer is no", explain how you get this conclusion (10 marks) (c) Find the Cartesian equation of the plane containing line PQ and the origin 0(0,0,0). 9 marks) Question 2 (20 marks) 1 2-i 4+3 (a) Gnen--+1 , evaluate z. (5 marks) b) The complex numbers z and have the following relationship 2- and Gven that the arguments ofa and :2 are within-? to +?. Find all possible zi and-2. (15 marks)

Explanation / Answer

PQ :
P + t(Q - P)
(1,2,-1) + t(4,-5,5)

PQ :
<1 + 4t , 2 - 5t , -1 + 5t> ---> ANS

Similarly MN :
M + s(N - M)

(1,4,6) + t(2,1,1)

MN :
<1 + 2s , 4 + s , 6 + s> ----> ANS

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b)
Lets see if they intersect, yo!

Equating x and y to find s and t :

PQ : <1 + 4t , 2 - 5t , -1 + 5t>
MN : <1 + 2s , 4 + s , 6 + s>

1 + 4t = 1 + 2s -----> 4t = 2s ----> S = 2t
2 - 5t = 4 + s

2 - 5t = 4 + 2t
7t = -2
t = -2/7

And s = 2t = -4/7

Now, check if the z's match....

PQ : z = -1 + 5t ---> -1 + 5(-2/7) ---> -1 - 10/7 ---> -17/7
MN : z = 6 + s ---> 6 + (-4/7) ---> 6 - 4/7 ----> 38/7

They dont match

So, no, these lines do not intersect

NO

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PQ = <1 + 4t , 2 - 5t , -1 + 5t>
and containing origin (0,0,0)

So, we have three points
P(1,2,-1)
Q(5,-3,4)
O(0,0,0)

direction vector of PQ = n1 = <4,-5,5>
direction vector of PO = n2 = <-1,-2,1>

Now, the normal vector of the plane is : n1 x n2 (cross product)

n = <5 , -9 , -13>
and point (0,0,0)

So, plane eqn is :
5(x - 0) + -9(y - 0) + -13(z - 0) = 0

5x - 9y - 13z = 0 ------> ANS

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