# Vector A has magnitude 3.7 units; vector B has magnitude 5.9. The angle between vector A and B is 45 degrees. What is the magnitude of vector A+ vector B?

Dec 7, 2017

Magnitude of $\left(\vec{A} + \vec{B}\right)$ is $8.91$ units at an angle
of
${27.92}^{0}$ from $\vec{A}$

#### Explanation:

Completing the parallelogram of $\vec{A} \mathmr{and} \vec{B}$ , the

diagonal represents represents resultant of two vectors

$\vec{A} \mathmr{and} \vec{B}$. The angle between $\vec{A}$and parallal

$\vec{B}$ is $180 - 45 = {135}^{0}$ By applying cosine law

we get magnitude of $| \vec{A} + \vec{B} {|}^{2}$

$= {3.7}^{2} + {5.9}^{2} - 2 \cdot 3.7 \cdot 5.9 \cdot \cos 135 \approx 79.37$ or

$| \vec{A} + \vec{B} | = 8.91$ units . Let the resultant vector.

$| \vec{A} + \vec{B} |$ makes an angle $\theta$ with $\vec{A}$.

By sine law $\frac{5.9}{\sin} \theta = \frac{8.91}{\sin} 135$ or

$\sin \theta = \frac{5.9 \cdot \sin 135}{8.91} \approx 0.4682$

or $\theta = {\sin}^{-} 1 \left(0.4682\right) \approx {27.92}^{0}$ from $\vec{A}$.

Magnitude of $\left(\vec{A} + \vec{B}\right)$ is $8.91$ units at an angle

of ${27.92}^{0}$ from $\vec{A}$ [Ans]