To find the dot product of the vectors [tex]\((-2, 7)\)[/tex] and [tex]\((-4, 7)\)[/tex], we follow these steps:
1. Identify the components of each vector:
- The first vector [tex]\(\mathbf{A}\)[/tex] has components [tex]\(A_x = -2\)[/tex] and [tex]\(A_y = 7\)[/tex].
- The second vector [tex]\(\mathbf{B}\)[/tex] has components [tex]\(B_x = -4\)[/tex] and [tex]\(B_y = 7\)[/tex].
2. Use the formula for the dot product of two vectors:
The dot product of vectors [tex]\(\mathbf{A} = (A_x, A_y)\)[/tex] and [tex]\(\mathbf{B} = (B_x, B_y)\)[/tex] is given by:
[tex]\[
\mathbf{A} \cdot \mathbf{B} = A_x \cdot B_x + A_y \cdot B_y
\][/tex]
3. Substitute the components into the formula:
- For [tex]\(A_x \cdot B_x\)[/tex], we have:
[tex]\[
-2 \cdot -4 = 8
\][/tex]
- For [tex]\(A_y \cdot B_y\)[/tex], we have:
[tex]\[
7 \cdot 7 = 49
\][/tex]
4. Add these products together:
[tex]\[
\mathbf{A} \cdot \mathbf{B} = 8 + 49
\][/tex]
[tex]\[
\mathbf{A} \cdot \mathbf{B} = 57
\][/tex]
Therefore, the dot product of the vectors [tex]\((-2, 7)\)[/tex] and [tex]\((-4, 7)\)[/tex] is [tex]\(57\)[/tex].
