To find the direction of the sum let's first find the sum, to do this we need to perform the vector decomposition:
[tex]\vec{A}=85\hat{i}[/tex]
and
[tex]\vec{B}=101\cos 60\hat{i}+101\sin 60\hat{j}[/tex]
Adding the vectors we have:
[tex]\begin{gathered} \vec{A}+\vec{B}=85\hat{i}+(101\cos 60\hat{i}+101\sin 60\hat{j}) \\ \vec{A}+\vec{B}=(85+101\cos 60)\hat{i}+101\sin 60\hat{j} \\ \vec{A}+\vec{B}=135.5\hat{i}+87.47\hat{j} \end{gathered}[/tex]
Now we need to remember that the angle of the vector is given by:
[tex]\theta=\tan ^{-1}(\frac{v_y}{v_x})[/tex]
where vx and vy are the x-component and y-component, respectively. Plugging the values we found we have that:
[tex]\begin{gathered} \theta=\tan ^{-1}(\frac{87.47}{135.5}) \\ \theta=32.84 \end{gathered}[/tex]
Therefore, the direction of the sum is 32.84°
