k: The Dot Question $$ 4, * 9.5 .11 $$ Part 1 of 3 For the following vectors, (a) find the dot product $$ \mathbf{v} \cdot \mathbf{w} $$; (b) find the angle between $$ \mathbf{v} $$ and $$ \mathbf{w} $$; (c) state whether the vectors are parallel, $$ 25 \%, 2 $$ of 8 points orthogonal, or neither. $$ \mathbf{v}=7 \mathbf{i}+4 \mathbf{j}, \mathbf{w}=4 \mathbf{i}-7 \mathbf{j} $$ (a) $$ \mathbf{v} \cdot \mathbf{w}=\square $$ (Simplify your answer.)

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To find the dot product $$ \mathbf{v} \cdot \mathbf{w} $$ for the vectors $$ \mathbf{v} = 7 \mathbf{i} + 4 \mathbf{j} $$ and $$ \mathbf{w} = 4 \mathbf{i} - 7 \mathbf{j} $$, we can use the formula for the dot product of two vectors:

$$
\mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2
$$

where $$ v_1 $$ and $$ v_2 $$ are the components of vector $$ \mathbf{v} $$, and $$ w_1 $$ and $$ w_2 $$ are the components of vector $$ \mathbf{w} $$.

Here, we have:
- $$ v_1 = 7 $$, $$ v_2 = 4 $$
- $$ w_1 = 4 $$, $$ w_2 = -7 $$

Now, we can calculate the dot product:

$$
\mathbf{v} \cdot \mathbf{w} = (7)(4) + (4)(-7)
$$

Calculating each term:

$$
= 28 - 28
$$

Thus, we find:

$$
\mathbf{v} \cdot \mathbf{w} = 0
$$

So, the answer for part (a) is:

$$
\mathbf{v} \cdot \mathbf{w} = 0
$$
$$\mathbf{v} \cdot \mathbf{w} = 0$$

k: The Dot Question
Part 1 of 3
For the following vectors, (a) find the dot product ; (b) find the angle between and ; © state whether the vectors are parallel, of 8 points
orthogonal, or neither.

(a) (Simplify your answer.)

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k: The Dot Question Part 1 of 3 For the following vectors, (a) find the dot product ; (b) find the angle between and ; © state whether the vectors are parallel, of 8 points orthogonal, or neither. (a) (Simplify your answer.)