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# vectors question

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1. The question is a line called L1, passes through the point A (2,0,2) and has the equation

R = (2,0,2) + T (2,6,-3)

The line L2 is parralel to this line and passes through the points B (4,4,-5) and C (0, -8, 1). P is the point on the line L1 such that angle CPA is a right angle, find the value of the parameter T at p.

i need to draw a sketch but am unsure how, im perplexted at this question any ideas? have I not made it clear enough?
2. (Original post by SunDun111)
The question is a line called L1, passes through the point A (2,0,2) and has the equation

R = (2,0,2) + T (2,6,-3)

The line L2 is parralel to this line and passes through the points B (4,4,-5) and C (0, -8, 1). P is the point on the line L1 such that angle CPA is a right angle, find the value of the parameter T at p.

i need to draw a sketch but am unsure how, im perplexted at this question any ideas? have I not made it clear enough?
Sketch doesn't have to be accurate.

Key point is you have two parallel lines. Mark B,C on one, and A on the other.
3. A diagram is helpful for the question simply to make it clearer to you what you need to do, but doesn't need to be accurate as already mentioned (and is by no means necessary once you are comfortable with this type of question).

Here's how to solve the question:
Since APC makes a right angle, you know that the scalar product of the vectors AP and CP is zero. You can find these vectors in terms of t and then simply multiply out the corresponding components (using the scalar product) to get a quadratic in t which can be easily solved (you know the quadratic equals 0 as I mentioned before).
Spoiler:
Show
t = -1, you'll also get t = 0 from the quadratic but this simply tells you that the zero vector dotted with vector CP is zero, which is true but not helpful.

By the way, to solve the next part of the question you simply need to find the magnitude of CP using your calculated value for t, as this perpendicular line between the two lines is the shortest distance.

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