Split vectors [tex]\vec A[/tex] and [tex]\vec B[/tex] into their horizontal and vertical components. [tex]\vec A[/tex] makes a total angle of 42.0 deg + 90 deg = 132.0 deg with the positive x-axis and has magnitude 16.0 units. Then the component form of [tex]\vec A[/tex] is
[tex]\vec A=A_x\,\vec\imath+A_y\,\vec\jmath=16.0\cos132.0^\circ\,\vec\imath+16.0\sin132.0^\circ\,\vec\jmath[/tex]
[tex]\implies\vec A=-10.7\,\vec\imath+11.9\,\vec\jmath[/tex]
[tex]\vec B[/tex] makes a total angle of 31.0 deg + 180 deg = 211.0 deg with the positive x-axis and has magnitude 7.00 units, so its component form is
[tex]\vec B=7.00\cos211.0^\circ\,\vec\imath+7.00\sin211.0^\circ\,\vec\jmath[/tex]
[tex]\implies\vec B=-6.00\,\vec\imath-3.61\,\vec\jmath[/tex]
Then
[tex]\vec C=\vec B-\vec A=(-6.00-(-10.7))\,\vec\imath+(-3.61-11.9)\,\vec\jmath[/tex]
[tex]\implies\vec C=4.7\,\vec\imath-15.5\,\vec\jmath[/tex]
The magnitude of [tex]\vec C[/tex] is
[tex]|\vec C|=\sqrt{4.7^2+(-15.5)^2}=16.2[/tex]
