Question

For vectors bold u equals negative 6 bold i plus 2 bold j and bold v equals 7 bold i minus 4 bold j, find left parenthesis negative bold u right parenthesis dot bold v.
(1 point)
Responses

negative 450
Image with alt text: negative 450

negative 50
Image with alt text: negative 50

108
108

150
150

Answer 1

To find \((- \mathbf{u}) \cdot \mathbf{v}\), we first need to compute \(-\mathbf{u}\):

Given
\[
\mathbf{u} = -6 \mathbf{i} + 2 \mathbf{j}
\]
we have
\[
-\mathbf{u} = -(-6 \mathbf{i} + 2 \mathbf{j}) = 6 \mathbf{i} - 2 \mathbf{j}
\]

Next, we compute the dot product \((- \mathbf{u}) \cdot \mathbf{v}\):

Given
\[
\mathbf{v} = 7 \mathbf{i} - 4 \mathbf{j}
\]

The dot product is defined as:
\[
(-\mathbf{u}) \cdot \mathbf{v} = (6 \mathbf{i} - 2 \mathbf{j}) \cdot (7 \mathbf{i} - 4 \mathbf{j})
\]
Using the formula for the dot product:
\[
\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2
\]
we find:
\[
(-\mathbf{u}) \cdot \mathbf{v} = (6)(7) + (-2)(-4)
\]
Calculating each term:
\[
= 42 + 8 = 50
\]

Therefore,
\[
(-\mathbf{u}) \cdot \mathbf{v} = 50
\]

Thus, the answer is **50**.