1. Subtract vector [tex]\( v = \langle 2, -3 \rangle \)[/tex] from vector [tex]\( u = \langle 5, 2 \rangle \)[/tex]:
[tex]\[
u - v = \langle 5, 2 \rangle - \langle 2, -3 \rangle
\][/tex]
Subtract the corresponding components:
[tex]\[
u - v = \langle 5 - 2, 2 + 3 \rangle = \langle 3, 5 \rangle
\][/tex]
2. Calculate the magnitude of the resulting vector [tex]\( \langle 3, 5 \rangle \)[/tex]:
The magnitude [tex]\( \|\langle 3, 5 \rangle\| \)[/tex] is found using the Pythagorean theorem:
[tex]\[
\|\langle 3, 5 \rangle\| = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.83
\][/tex]
3. Calculate the angle of direction of the vector [tex]\( \langle 3, 5 \rangle \)[/tex]:
The angle of direction [tex]\( \theta \)[/tex] is found using the tangent function:
[tex]\[
\theta = \arctan\left(\frac{5}{3}\right)
\][/tex]
To convert this angle from radians to degrees:
[tex]\[
\theta_{\text{degrees}} = \arctan\left(\frac{5}{3}\right) \times \frac{180}{\pi} \approx 59.04^\circ
\][/tex]
Therefore, the magnitude of the resulting vector, [tex]\( \langle 3, 5 \rangle \)[/tex], is approximately 5.83, and its angle of direction is approximately 59.04 degrees.
