Question

Determine Dot products and angle between vectors.

a) $u·v$

b) The angle between u and v to the nearest degree.

Short answer

a. $u·v=2$

b. $\theta =45°$

Step-by-step solution

a.Step 1. Given information.

$\begin{array}{l}u=〈2,0〉\\ v=〈1,1〉\end{array}$

Step 2. Determining the dot product.

The dot product of vectors

$\overrightarrow{u} .\overrightarrow{v} =a_1{b}_1+a_2{b}_2$

$\overrightarrow{u} .\overrightarrow{v} =(2\times 1)+(0\times 1)$

$\overrightarrow{u} .\overrightarrow{v} =2+0$

$\overrightarrow{u} .\overrightarrow{v} =2$

b.Step 1. Given information.

$\begin{array}{l}u=〈2,0〉\\ v=〈1,1〉\end{array}$

also from part (a) $\overrightarrow{u} ·\overrightarrow{v} =2$

Now, $\cos \theta =\frac{\overrightarrow{u} ·\overrightarrow{v} }{|\overrightarrow{u} ||\overrightarrow{v} |}$

$|\overrightarrow{u} |=\sqrt{2^2+0^2}$

$|\overrightarrow{u} |=\sqrt{4}$

$|\overrightarrow{u} |=2$

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

$|\overrightarrow{v} |=\sqrt{2}$

Hence,

$\cos \theta =\frac{2}{2\sqrt{2}}$

$\cos \theta =\frac{1}{\sqrt{2}}$

$\theta =45°$