Parametric equations!

Question

Q1a) Let x = p + td be a parametric equation of the line through the points A(4,4,1)  and B(7,0,0)

i) What is the vector p, given that its second component is 4?

ii) Give the vector d that has the first component 3

 

b) Let x = q + se be a parametric equation of the line through the point C(-3,1,4) and in the direction (-6,-2,5)

i) What is the vector q, given that its third component is 4?

ii) Give the vector e that has second component -2

 

c) Are these two lines the same?

Yes or No

 

d) Is the line with parametric equation x = (1,0,1) + r(-1,-3,9) perpendicular to the line AB?

 

Just need help with c) and d) but not too sure if all points in a) and b) are needed. Thank you.

Answer 1

 

 

 

To decide if the lines are coincident, we check to see whether a point on one of the lines satisfies the vector equation of the other. The point (- 3, 1, 4) is on the second line. If it is also on the first line, then (- 3, 1, 4) = (4, 4, 1) + t(3, - 4, - 1).

Then - 3 = 4 + 3t                1 = 4 - 4t                4 = 1 - t

            3t = - 7                     4t = 3                     t = - 3

           t = - 7/3                   t = 3/4                     t = - 3

Since the parameter value t is different obtained each equation, the point (- 3, 1, 4) from the second line does not lie on the first line, and the lines are distinct.

The two line are not parallel and these are distinct. So, the two lines are not same.