Linear independence and vector parametrics!

  • #1

Homework Statement


Consider the two lines, given in the paramentric form

L1: x = (0, 1 ,2) + s(1, 0, 2)
L2: x = (4, 2, c) + t(-2, 0, d)

where c and d are constants.

a) For what value of d are the lines parallel?
b) With the value of d above, for what value(s) of c (if any) are the two lines identical? Justify briefly.
c) For the case c = 5 and d = 0 find the point P on L1 and Q on L2 such that the distance between P and Q is as small as possible.

2. A concept question:
Can any vector x with two components be expressed as a linear combination of a and b? Why?

Homework Equations



?

The Attempt at a Solution


a) I think d = -4, but not sure if that's right.
b) I'm not sure how to approach this problem. Do I make 2 + 2s = c - 4t
But what do I do after that?
c) No clue. I'm thinking perhaps using the dot product = 0?

2. I think a vector can only be expressed as a linear combination of a and b if the two vectors are linearly independent. How do I prove that?
 

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