Dot Product and vector length

Question

Hi! I am working on some online homework for my calc2 class that covers the dot product and I am really struggling with this one question. I understood how to solve part a, because we covered that in class, but I am not sure how to solve parts b and c. I tried plugging in the values that were provided for ||v||and ||w|| into the equations of parts b and c, but that did not work. If someone can help me solve parts b and c of this problem I would really appreciate it!

Answer 1

Hint: If $\mathbf{a}$ and $\mathbf{b}$ are vectors and $\alpha$ and $\beta$ are scalars then \begin{align} \left\|\alpha\mathbf{a}+\beta\mathbf{b}\right\|^2&=(\alpha\mathbf{a}+\beta\mathbf{b})\cdot (\alpha\mathbf{a}+\beta\mathbf{b})\\ &=\alpha^2\left\|\mathbf{a}\right\|^2+2\alpha\beta\mathbf{a}\cdot\mathbf{b}+\beta^2\left\|\mathbf{b}\right\|^2\\ &=\alpha^2\left\|\mathbf{a}\right\|^2+2\alpha\beta\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\cos\color{red}{\theta}+\beta^2\left\|\mathbf{b}\right\|^2 \end{align} Where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.

Answer 2

Note that $$||a+b||^2=||a||^2+||b||^2$$ Hope this helps,

Answer 3

Simplest way: WLoG, let $w=3\hat{x}$