Question

problem 6
(fill in the blank) : given the vectors,
\\( \vec{v} = 3\hat{i} - 2\hat{j} + \hat{k} \\)
and
\\( \vec{w} = -2\hat{i} + 3\hat{j} + 7\hat{k} \\),
please calculate the dot product \\( \vec{v} \cdot \vec{w} \\).
\\( \vec{v} \cdot \vec{w} = \text{<your answer here>}

problem 7
(fill in the blank) : given the dot products,
\\( \vec{v} \cdot \vec{u} = 5 \\)
and
\\( \vec{w} \cdot \vec{u} = 8 \\),
please evaluate the expression, \\( \vec{u} \cdot (3\vec{v} - \vec{w}) \\).
\\( \vec{u} \cdot (3\vec{v} - \vec{w}) = \text{<your answer here>}

Explanation:

Problem 6

Step1: Recall dot product formula

For $\vec{v}=v_1\hat{i}+v_2\hat{j}+v_3\hat{k}$ and $\vec{w}=w_1\hat{i}+w_2\hat{j}+w_3\hat{k}$, $\vec{v}\cdot\vec{w}=v_1w_1+v_2w_2+v_3w_3$

Step2: Substitute vector components

$\vec{v}\cdot\vec{w}=(3)(-2)+(-2)(3)+(1)(7)$

Step3: Compute each term and sum

$\vec{v}\cdot\vec{w}=-6-6+7$

Problem 7

Step1: Use dot product distributive property

$\vec{u}\cdot(3\vec{v}-\vec{w})=3(\vec{u}\cdot\vec{v})-(\vec{u}\cdot\vec{w})$

Step2: Substitute given dot product values

$\vec{u}\cdot(3\vec{v}-\vec{w})=3(5)-8$

Step3: Calculate the result

$\vec{u}\cdot(3\vec{v}-\vec{w})=15-8$

Answer:

Problem 6: $\vec{v} \cdot \vec{w} = -5$
Problem 7: $\vec{u} \cdot (3\vec{v} - \vec{w}) = 7$