# Calculate |vec u + vec v| help?

## Given that |$\vec{u}$| = 3 and |$\vec{v}$| = 4 ALSO calculate the angle between $\vec{u}$ and $\vec{u}$ + $\vec{v}$

Jun 9, 2018

See below

#### Explanation:

First Part

• $\boldsymbol{u} \cdot \boldsymbol{v} = \left\mid \boldsymbol{u} \right\mid \left\mid \boldsymbol{v} \right\mid \cos \phi$

$\left\mid \boldsymbol{u} \right\mid = 3 , \left\mid \boldsymbol{v} \right\mid = 4 , \phi = \frac{2 \pi}{3}$

${\left\mid \boldsymbol{u} + \boldsymbol{v} \right\mid}^{2} = \left(\boldsymbol{u} + \boldsymbol{v}\right) \cdot \left(\boldsymbol{u} + \boldsymbol{v}\right)$

$= {u}^{2} + {v}^{2} + 2 \boldsymbol{u} \cdot \boldsymbol{v}$

$= {3}^{2} + {4}^{2} + 2 \cdot 3 \cdot 4 \cdot \cos \left(120\right)$

$= 13$

$\implies \left\mid \boldsymbol{u} + \boldsymbol{v} \right\mid = \sqrt{13}$

Second Part

$\boldsymbol{u} \cdot \left(\boldsymbol{u} + \boldsymbol{v}\right) = \left\mid \boldsymbol{u} \right\mid \cdot \left\mid \boldsymbol{u} + \boldsymbol{v} \right\mid \cos \theta$

$\cos \theta = \frac{\boldsymbol{u} \cdot \left(\boldsymbol{u} + \boldsymbol{v}\right)}{\left\mid \boldsymbol{u} \right\mid \cdot \left\mid \boldsymbol{u} + \boldsymbol{v} \right\mid}$

$= \frac{{u}^{2} + \left\mid \boldsymbol{u} \right\mid \left\mid \boldsymbol{v} \right\mid \cos \phi}{\left\mid \boldsymbol{u} \right\mid \cdot \left\mid \boldsymbol{u} + \boldsymbol{v} \right\mid}$

$= \frac{{3}^{2} + 3 \cdot 4 \cdot \cos \left(120\right)}{3 \cdot \sqrt{13}}$

$= \frac{1}{\sqrt{13}}$

$\theta = {\cos}^{- 1} \left(\frac{1}{\sqrt{13}}\right) = {73.9}^{o}$