Two vectors \( v = \langle 1, 3 \rangle \) and \( w = \langle 2, 2 \rangle \) are given.
(a) What angle does v make with the x-axis?
(b) Find a vector u such that u + v = -w
The angle between a vector and the x-axis is the gradient of the vector
If \( \theta \) is the angle then
\[ \begin{align}
\tan \theta &= \frac{v_2}{v_1} \\
&= \frac{3}{1}
\end{align} \ \]
and the angle is
\[ \begin{align}
\theta &= \tan^{-1}(\frac{3}{1}) \\
&= 1.25 \quad \text{radians}
\end{align} \ \]
Use components
\[ \begin{align}
u + v &= -w \\
\end{align} \ \]
in components this is
\[ \begin{align}
\langle u_1, u_2 \rangle + \langle 1, 3 \rangle &= -\langle 2, 2 \rangle \\
\end{align} \ \]
Solve for the components of u
\[ \begin{align}
u_1 + 1 &= -2 \\
u_1 &= -3
\end{align} \ \]
and
\[ \begin{align}
u_2 + 3 &= -2 \\
u_2 &= -5
\end{align} \ \]
then
\[ \begin{align}
u &= \langle -3, -5 \rangle\\
\end{align} \ \]