2
$\begingroup$

Consider two nonzero vectors $a$ and $b$.

First, if $|a+b|=|a-b|$, how can I show that $a$ and $b$ are perpendicular?

And how can I show that $|a\times b|^2 = |a|^2 |b|^2 - (a\cdot b)^2$ ?

$a\cdot b$ is the dot product.

  • 0
    Hint for first part: $|a+b|^2=(a+b)\cdot(a+b)=a\cdot a+2(a\cdot b)+b\cdot b$.2012-04-08

1 Answers 1

1

For the first, square both sides and see what cancels out. If two vectors are perpendicular, what is their dot product?

For the second, remember that $a \cdot b = \|a\|\|b\|\cos{\theta}$. Plug in and use a trigonometric identity to change $\cos$ to $\sin$.