# Thread: Vector proof

1. ## Vector proof

Show that vectors (b dot c)a - (a dot c)b and vector c are perpendicular. Any advice is appreciated. Frostking

2. Originally Posted by Frostking
Show that vectors (b dot c)a - (a dot c)b and vector c are perpendicular. Any advice is appreciated. Frostking
Let $\mathbf{z}=\left(\mathbf{b}\cdot\mathbf{c}\right)\ mathbf{a}-\left(\mathbf{a}\cdot\mathbf{c}\right)\mathbf{b}$

Thus, you want to show that $\mathbf{z}\cdot\mathbf{c}=0$

I'll start you off and then I'll let you finish

$\mathbf{z}\cdot\mathbf{c}=\left[\left(\mathbf{b}\cdot\mathbf{c}\right)\mathbf{a}\r ight]\cdot\mathbf{c}-\left[\left(\mathbf{a}\cdot\mathbf{c}\right)\mathbf{b}\r ight]\cdot\mathbf{c}=\left(\mathbf{b}\cdot\mathbf{c}\ri ght)\left(\mathbf{a}\cdot\mathbf{c}\right)-\left(\mathbf{a}\cdot\mathbf{c}\right)\left(\mathb f{b}\cdot\mathbf{c}\right)=\dots$

Does this make sense?

3. It helps to know that $C \times \left( {A \times B} \right) = \left( {C \cdot B} \right)A - \left( {C \cdot A} \right)B$
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