<s>[INST] Vectors Year 11 
This is a vectors question I got, could any confirm my answer? I got $6.066\underline i - 279.28\underline j$.
$AB = \underline B  - \underline A $
(Working on Vector $\underline B$) 
\begin{align}x &= |B|\cos \theta \\
x &= 150\cos 315 \\
x &= 75\sqrt 2 \end{align}
\begin{align}y &= |B|\sin \theta \\
y &= 150\sin 315 \\
y &= -75\sqrt 2 \end{align}
(Working on Vector $\underline A$)
\begin{align}x = |A|\cos \theta \\
x = 200\cos 60 \\
x = 100 \end{align}
\begin{align}y &= |A|\sin \theta \\
y &= 200\sin 60 \\
y &= -173.21 \text{(2 decimals)} \end{align}
$\underline B  - \underline A  = 75\sqrt 2\underline i - 75\sqrt 2\underline j + -(100\underline i + 173.21\underline j)$ 
$\underline B  - \underline A  = 6.066\underline i - 279.28\underline j$
 [/INST] The top vector has angle $\frac \pi 3$ above the horizontal, and so its components are
$$\vec A = 200\cos\frac\pi 3\hat{i}+200\sin \frac \pi 3 \hat j$$
Similarly, the angle for the other vector is $-\frac \pi 4$, so it is
$$\vec B = 150\cos\frac{-\pi} 4\hat{i}+150\sin \frac {-\pi} 4 \hat j$$
The resulting vector is
$$\vec A+\vec B = \left(200\cos \frac\pi 3 +150\cos\frac \pi 4\right)\hat i+\left(200\sin\frac\pi 3-150\sin\frac\pi 4\right)$$
$$\approx 206\hat i-67.1\hat j$$
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