Vector Questions?

Question

you don't have to help me on the 2 questions, just one im happy enough, the thing is i do not understand vectors much, so theres no point of getting answers without understanding, but if you can do at least one question, with your knowledge i would be grateful :) thanks!!



1.
Prove that the line x = 1 + 2t, y = -1 + 3t, z = 2 + 4t is parallel to the plane x - 2y + z = 6. What is the distance between the line and the plane?

2.

Let A, B and C be three non-parallel vectors in R3. Express A x (B x C)
in the form α*B + β*C with scalars α and β :
(i) Explain why it is clear that A x (B x C) must be a linear combination
of B and C;
(ii) Choose an orthonormal basis ijk adapted to the problem and prove it by
standard properties of the vector product.
(Note: there is a way of choosing the orthonormal basis in a way which makes
all the calculations very easy

Answer 1

The direction of the line is <2,3,4>. The normal to the plane is <1,-2,1>. The dot product of the two vectors is 2*1 + 3*(-2) + 4*1 = 0. This says that the two vectors are orthogonal (which is an advanced way of saying they are perpendicular). Since the normal is orthogonal to the line, the line must be parallel to the plane.