# Finding Vectors from their angle and magnitude

The question is as follows:

I started to approach this by thinking of the vectors $$v$$ and $$w$$ as of the form $$a \hat i+b \hat j$$ and $$c \hat i+d \hat j$$. Based on this, I created three equations:

$$\sqrt{a^2+b^2} = 3$$ and $$\sqrt{c^2+d^2} = \sqrt{2}$$ from the vector magnitudes that are given, and $$ac+bd=3$$ from the vector angle equation $$\hat u \cdot \hat v = |\hat u||\hat v|cos( \theta)$$.

However, as there are four variables, and only three equations, I am obviously missing something here in order to solve the problem. What vector equation am I missing to be able to determine the vectors, and therefore this arbitrary magnitude which the question asks for?

• You don't need to know all the variables to solve the magnitude. Just expand the sum $|v +2w|^2 = (v+2w) \cdot (v+2w)$. The info you're given should be enough to solve it. Commented Aug 22, 2021 at 18:36
• Another method: if you draw the two vectors $\ v \$ and $\ 2w \$ head-to-tail and then draw in their sum, you will form a triangle with the included angle between the two given vectors being $\ \pi - \frac{\pi}{4} \ = \ \frac{\ 3 \pi}{4} \ \ .$ You will be able to get the length-squared of the sum-vector $\ v + 2w \$ from the Law of Cosines.
– user882145
Commented Aug 22, 2021 at 20:12

You have\begin{align}|\mathbf v+2\mathbf w|^2&=|\mathbf v|^2+4|\mathbf w|^2+4\langle\mathbf v,\mathbf w\rangle\\&=17+4\langle\mathbf v,\mathbf w\rangle.\end{align}Furthermore, since the angle between $$\mathbf v$$ and $$\mathbf w$$ is $$\frac\pi4$$,$$\langle\mathbf v,\mathbf w\rangle=|\mathbf v||\mathbf w|\cos\left(\frac\pi4\right)=3.$$Therefore, $$|\mathbf v+2\mathbf w|^2=29$$.
• Thanks! Shouldn't $|v|^2+4|w|^2$ be 17 though, and the answer would be 29? Commented Aug 22, 2021 at 18:46