I am working on my maths homework and encounter the following question which I have no clue to answer:
Let A = (1, 1, 2), B = (-3, 1, 4), C = (-1, -1, 0) be points in space.
Q1: Find all values x such that the vector u = [2, 14, x] is a linear combination of vectors AB and AC.
I got vectors AB [ -4, 0, 2] and AC [-2, -2, -2], and combination of vectors AB and AC [-6, -2, 0]. Then I equated that with vector u in the following manner:
[2, 14, x] = a[-4, 0, 2] + b[-2, -2, -2] where a = 3 and b = -7 which gives x = 20. Is this the only value of x?
Q2: Find a vector form and parametric equations for the line passing through A and perpendicular to the triangle ABC.
This one I surrender. No clue.
Please kindly help cos I have been stuck in this question for hours. Many many thanks!!!
Q1 Yes, it's correct (unless you miscalculated), $x=10$ is the only value that makes it linearly dependent to $AB$ and $AC$.
Q2 $AB\times AC$ gives a normal vector to $ABC$ plane, so it's the direction vector of the line perpendicular to this plane, and you also know a point on the line, namely $A$.