To find the dot product of two vectors [tex]\(\mathbf{a}\)[/tex] and [tex]\(\mathbf{b}\)[/tex], you can use the formula:
[tex]\[
\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3
\][/tex]
where [tex]\(\mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k}\)[/tex] and [tex]\(\mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k}\)[/tex].
Given:
[tex]\[
\mathbf{a} = -5 \mathbf{i} + 2 \mathbf{j} + 7 \mathbf{k}
\][/tex]
[tex]\[
\mathbf{b} = 4 \mathbf{i} + 3 \mathbf{j} + 7 \mathbf{k}
\][/tex]
We identify the components:
[tex]\[
a_1 = -5, \quad a_2 = 2, \quad a_3 = 7
\][/tex]
[tex]\[
b_1 = 4, \quad b_2 = 3, \quad b_3 = 7
\][/tex]
Now, apply these values to the dot product formula:
[tex]\[
\mathbf{a} \cdot \mathbf{b} = (-5)(4) + (2)(3) + (7)(7)
\][/tex]
Calculate each term:
[tex]\[
-5 \times 4 = -20
\][/tex]
[tex]\[
2 \times 3 = 6
\][/tex]
[tex]\[
7 \times 7 = 49
\][/tex]
Add these results together:
[tex]\[
-20 + 6 + 49
\][/tex]
Combine the sums:
[tex]\[
-20 + 6 = -14
\][/tex]
[tex]\[
-14 + 49 = 35
\][/tex]
Therefore, the dot product of the vectors [tex]\(\mathbf{a}\)[/tex] and [tex]\(\mathbf{b}\)[/tex] is:
[tex]\[
35
\][/tex]
