Prove that these two vectors are orthogonal

||b||a + ||a||b and ||b||a - ||a||b

Here's my attempt:

Since I wish to show that these vectors are orthogonal, I will attempt to show that the dot product must be 0.

(||b||a + ||a||b) * (||b||a - ||a||b)

= ||b||(a * a) + (||b||a * ||a||b) + (||a||b * ||b||a) + ||a||(b * -b)

= ||b||||a||^2 + ||b||^2||a||^2 - ||b||^2||a||^2 + ||a||||b||^2(-1)

= ||b||||a||^2 - ||a||||b||^2

But I see nothing to suggest that this is 0.

Hints?

Nevermind, I made a careless error and that last line should have all magnitudes squared.

Quote:

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Originally Posted by OneMileCrash

||b||a + ||a||b and ||b||a - ||a||b

Here's my attempt:

Since I wish to show that these vectors are orthogonal, I will attempt to show that the dot product must be 0.

(||b||a + ||a||b) * (||b||a - ||a||b)

= ||b||(a * a) + (||b||a * ||a||b) + (||a||b * ||b||a) + ||a||(b * -b)

= ||b||||a||^2 + ||b||^2||a||^2 - ||b||^2||a||^2 + ||a||||b||^2(-1)

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Line 3 should read (after canceling the middle terms)
= ||b||^2 * ||a||^2 - ||a||^2 * ||b||^2

-Dan