Showing that two vectors are perpendicular

gmer

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Feb 11, 2012
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Regarding the problem:

Show that for any vectors \(\displaystyle \vec{a}\), \(\displaystyle \vec{b}\in{R^3}\),
\(\displaystyle \vec{u} = |\vec{b}|\vec{a} + |\vec{a}|\vec{b}\) and \(\displaystyle \vec{v} = |\vec{b}|\vec{a} - |\vec{a}|\vec{b}\)
are perpendicular

I'm not exactly sure where to begin with this problem. I think maybe taking the dot product of u and v but I'm not sure.
 
Show that for any vectors \(\displaystyle \vec{a}\), \(\displaystyle \vec{b}\in{R^3}\),
\(\displaystyle \vec{u} = |\vec{b}|\vec{a} + |\vec{a}|\vec{b}\) and \(\displaystyle \vec{v} = |\vec{b}|\vec{a} - |\vec{a}|\vec{b}\)
are perpendicular

I'm not exactly sure where to begin with this problem. I think maybe taking the dot product of u and v but I'm not sure.
What does \(\displaystyle \vec{u}\cdot\vec{v}=~?\)
Hint: Recall that \(\displaystyle \vec{a}\cdot\vec{a}=\|a\|^2.\)
 
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