Question

Question #ea7e8

Answer 1

We have vectors `bb(ul a)` and `bb(ul b)` such that:

`(bb(ul a) * bb(ul b))^2 = | bb(ul a) |^2 \ | bb(ul b) |^2`

Using the definition of the dot (or scalar) product, and denoting the angle between `bb(ul a)` and `bb(ul b)` by `theta`, we have:

`(| bb(ul a) | \ | bb(ul b) | \ cos theta)^2 = | bb(ul a) |^2 \ |bb(ul b)|^2`

And so:

`| bb(ul a) |^2 \ |bb(ul b)|^2 - | bb(ul a) |^2 \ | bb(ul b) |^2 \ cos^2theta = 0`
`:. | bb(ul a) |^2 \ |bb(ul b) |^2 \ ( 1 - cos theta )= 0`

Leading to:

A) Either: `| bb(ul a) |^2 \ |bb(ul b) |^2 => | bb(ul a) | = 0` or `| bb(ul b) | = 0`

`=> bb(ul a) = 0` or `bb(ul b) = 0`

But we are given that `bb(ul a)` and `bb(ul b)` are non-zero, eliminating this as as possible solution.

B) Or: `cos theta-1=0=> cos theta =1`

`=> theta =0`
`=> bb(ul a)` and `bb(ul b)` are parallel QED.