To determine if we can add or subtract the given vectors, we need to ensure that both vectors have the same dimensions. Given two vectors [tex]\( \mathbf{a} \)[/tex] and [tex]\( \mathbf{b} \)[/tex], vector addition [tex]\( \mathbf{a} + \mathbf{b} \)[/tex] or subtraction [tex]\( \mathbf{a} - \mathbf{b} \)[/tex] can only be performed if:
1. Both vectors have the same number of components, meaning the same length.
Let's look at each part of this problem.
The first part of the expression is:
[tex]\[
\left[ \begin{array}{llll}
5 & 2 & -1 & 3
\end{array} \right]
+
\left[ \begin{array}{llll}
1 & -6 & -6 & -3
\end{array} \right]
\][/tex]
Both vectors have the same number of components (4 each). So, we can add them component-wise:
[tex]\[
\left[5 + 1, 2 + (-6), -1 + (-6), 3 + (-3)\right]
=
\left[6, -4, -7, 0\right]
\][/tex]
This results in a new vector:
[tex]\[
\left[ \begin{array}{llll}
6 & -4 & -7 & 0
\end{array} \right]
\][/tex]
Now, the problem asks to compare this result to the given vectors:
[tex]\[
\begin{array}{llll}
\left[ \begin{array}{llll}
6 & -4 & 5 & 6
\end{array} \right]
\\
\left[ \begin{array}{llll}
4 & 8 & 5 & 6
\end{array} \right]
\\
\left[ \begin{array}{llll}
6 & -4 & -7 & 0
\end{array} \right]
\end{array}
\][/tex]
The calculated vector [tex]\(\left[6, -4, -7, 0\right]\)[/tex] is:
- Not equal to [tex]\(\left[6, -4, 5, 6\right]\)[/tex]
- Not equal to [tex]\(\left[4, 8, 5, 6\right]\)[/tex]
- But is equal to [tex]\(\left[6, -4, -7, 0\right]\)[/tex]
Thus, the last given vector matches our resulting vector.
In summary:
- The vectors [tex]\(\left[5, 2, -1, 3\right]\)[/tex] and [tex]\(\left[1, -6, -6, -3\right]\)[/tex] can indeed be added.
- The correct resultant vector is [tex]\(\left[6, -4, -7, 0\right]\)[/tex].
