Question

In the session 10 problems, 1b, we use the dot product to get the angle between Av and v. In the solutions, I don't understand how they get that (Av dot v) / |Av||v| =1/sqrt(2).

I understand that the length is sqrt(x^2+y^2) and the dot product is (x^2+y^2)/sqrt(2).

Answer 1

\[\begin{aligned}
Av\cdot v &= |Av||v|\cos\theta \\
\therefore \cos\theta &= \frac{\color{blue}{Av\cdot v}}{\color{red}{|Av|}\color{teal}{|v|}} \\
&= \color{blue}{\frac{x^2+y^2}{\sqrt{2}}} \cdot \frac{1} {\color{red}{\sqrt{x^2+y^2}}\color{teal}{\sqrt{x^2+y^2}}} \\
&= \frac{x^2+y^2}{\sqrt{2}} \cdot \frac{1} {x^2+y^2} \\
&= {1 \over \sqrt{2}}
\end{aligned}\]

Answer 2

Oh! thanks.