# How do you find the angle between the vectors u=5i+5j and v=-6i+6j?

Nov 2, 2016

Please see the explanation to understand why we know that the angle between the two vectors is $\frac{\pi}{2}$

#### Explanation:

Given two vectors of the form:

$\overline{a} = {a}_{\hat{i}} \hat{i} + {a}_{\hat{j}} \hat{j}$

and

$\overline{b} = {b}_{\hat{i}} \hat{i} + {b}_{\hat{j}} \hat{j}$

The dot-product is:

$\overline{a} \cdot \overline{b} = \left({a}_{\hat{i}}\right) \left({b}_{\hat{i}}\right) + \left({a}_{\hat{j}}\right) \left({b}_{\hat{j}}\right)$

Therefore, the dot-product of the two vectors:

$\overline{u} = 5 \hat{i} + 5 \hat{j}$ and $\overline{v} = - 6 \hat{i} + 6 \hat{j}$ is

$\overline{u} \cdot \overline{v} = \left(5\right) \left(- 6\right) + \left(5\right) \left(6\right) = 0$

Normally, we would use the above dot-product the for the left side of the equation:

$\overline{u} \cdot \overline{v} = | \overline{u} | | \overline{v} | \cos \left(\theta\right)$

And then compute the magnitudes and, ultimately, solve for $\theta$.

But, because the dot-product is zero and we know that the magnitudes cannot be zero, then we know that:

$\cos \left(\theta\right) = 0$

Which means that:

$\theta = \frac{\pi}{2}$