Question

For vectors u= (3,4) and v= (1,3) find CompuV and the angle between u and v.

Answer 1

Answer:

The angle between the given vectors u and v is $\theta=cos^{-1}\left[\frac{3}{\sqrt{10}}\right]$

Step-by-step explanation:

Given vectors are $\overrightarrow{u}=(3,4)$ and $\overrightarrow{v}=(1,3)$

Now compute the dot product of u and v:

$\overrightarrow{u}.\overrightarrow{v}=(3,4).(1,3)$

  $=(3)(1)+(4)(3)$

  $=3+12$

 $=15$

Now find the magnitude of u and v:

$|\overrightarrow{u}|=\sqrt{3^2+4^2}$

$=\sqrt{9+16}$

$=\sqrt{25}$

$=5$

$|\overrightarrow{u}|=5$

$|\overrightarrow{v}|=\sqrt{1^2+3^2}$

$=\sqrt{1+9}$

$=\sqrt{10}$

$|\overrightarrow{v}|=\sqrt{10}$

To find the angle between the given vectors

$\overrightarrow{u}.\overrightarrow{v}=|\overrightarrow{u}|\overrightarrow{v}|cos\theta$

$\theta=cos^{-1}\left[\frac{\overrightarrow{u}.\overrightarrow{v}}{|\overrightarrow{u}|\overrightarrow{v}|}\right]$

$=cos^{-1}\left[\frac{15}{5\times \sqrt{10}}\right]$

$=cos^{-1}\left[\frac{15}{5\times \sqrt{10}}\right]$

$\theta=cos^{-1}\left[\frac{3}{\sqrt{10}}\right]$

Therefore the angle between the vectors u and v is

$\theta=cos^{-1}\left[\frac{3}{\sqrt{10}}\right]$