A mathematical problem (vectors)?

Question

A and B have position vectors (3i + 6j -8k) and (8j - 6k) respectively.

The line L1 passes through both these, and I have found its equation as (3i + 6j -8k) + s(-3i+2k+2j)

The line L2 has equation (-2i + 10j + 6k) + t(7i - 4j + 6k)

I have found that these lines intersect at (-9i + 14j) and s=4, t=-1

The point C lies on L2 such that AC is perpendicular to AB. What do I need to use to find the position vector of C - I think it's the dot product/scalar product rule but I'm not sure, and if it is, which vectors do I need to use?

Answer 1

You have OC (I can't do the over-arrows - please read them as present, along with underlines for i, j & k!) in terms of t.

So you can AC in terms of t since AC = OC - OA

You have also found AB = -3i + 6j - 8k

Find the dot product of AC with AB. ie multiply together the respective multipliers of i, j and k and add the three together. Set this equal to 0 (the condition for two vectors to be perpendicular). This will give you an equation in t which you can then solve and hence find OC.

Hope this helps

Answer 2

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Answer 3

dot product