Question

given \\( \mathbf{v} = 5\mathbf{i} + \mathbf{j} \\) and \\( \mathbf{w} = \mathbf{i} + 5\mathbf{j} \\), find the following.\
a) \\( \mathbf{v} \cdot \mathbf{w} \\)\
b) \\( \mathbf{v} \cdot \mathbf{v} \\)\
\
a) \\( \mathbf{v} \cdot \mathbf{w} = \square \\)

Explanation:

Step1: Recall dot product formula

For vectors $\mathbf{v}=a_1\mathbf{i}+b_1\mathbf{j}$ and $\mathbf{w}=a_2\mathbf{i}+b_2\mathbf{j}$, $\mathbf{v}\cdot\mathbf{w}=a_1a_2 + b_1b_2$.

Step2: Calculate $\mathbf{v}\cdot\mathbf{w}$

Substitute $a_1=5, b_1=1, a_2=1, b_2=5$:
$\mathbf{v}\cdot\mathbf{w}=(5)(1)+(1)(5)=5+5$

Step3: Calculate $\mathbf{v}\cdot\mathbf{v}$

Substitute $a_1=5, b_1=1, a_2=5, b_2=1$:
$\mathbf{v}\cdot\mathbf{v}=(5)(5)+(1)(1)=25+1$

Answer:

a) $\mathbf{v}\cdot\mathbf{w}=10$
b) $\mathbf{v}\cdot\mathbf{v}=26$