How do you compute the dot product for u=i-2j and v=-2i+j?

Nov 10, 2016

$\therefore \underline{u} \cdot \underline{v} = - 4$

Explanation:

Inner Product Definition
If $\underline{u} = \left\langle\left({u}_{1} , {u}_{2}\right)\right\rangle$, and $\underline{v} = \left\langle\left({v}_{1} , {v}_{2}\right)\right\rangle$, then the inner product (or dot product), a scaler quantity, is given by:
$\underline{u} \cdot \underline{v} = {u}_{1} {v}_{1} + {u}_{2} {v}_{2}$

Inner Product = 0 $\Leftrightarrow$ vectors are perpendicular

So, $\underline{u} = \underline{\hat{i}} - 2 \underline{\hat{j}}$, and $\underline{v} = - 2 \underline{\hat{i}} + \underline{\hat{j}}$

Then the inner product is given by;
$\underline{u} \cdot \underline{v} = \left(1\right) \left(- 2\right) + \left(- 2\right) \left(1\right)$
$\therefore \underline{u} \cdot \underline{v} = - 2 - 2$
$\therefore \underline{u} \cdot \underline{v} = - 4$