given ( v = 5i + j ) and ( w = i + 5j ), find the following.\na) ( vcdot w ) b) ( vcdot v )\na) ( vcdot…

Explanation:

Step1: Recall the dot - product formula

If (\vec{v}=a_1\vec{i}+b_1\vec{j}) and (\vec{w}=a_2\vec{i}+b_2\vec{j}), then (\vec{v}\cdot\vec{w}=a_1a_2 + b_1b_2).

Step2: Calculate (\vec{v}\cdot\vec{w})

Given (\vec{v}=5\vec{i}+\vec{j}) ((a_1 = 5), (b_1=1)) and (\vec{w}=\vec{i}+5\vec{j}) ((a_2 = 1), (b_2 = 5)). [ \begin{align*} \vec{v}\cdot\vec{w}&=(5\times1)+(1\times5)\ &=5 + 5\ &=10 \end{align*} ]

Step3: Calculate (\vec{v}\cdot\vec{v})

Here (a_1 = 5), (b_1=1), (a_2 = 5), (b_2 = 1) (since (\vec{v}) is dotted with itself). [ \begin{align*} \vec{v}\cdot\vec{v}&=(5\times5)+(1\times1)\ &=25+1\ &=26 \end{align*} ]

Answer:

a) (10) b) (26)