Question

What is the sum of the vectors \( \begin{bmatrix} 4 \\ -5 \end{bmatrix} \) and \( \begin{bmatrix} 10 \\ 2 \end{bmatrix} \)?

a. \( \begin{bmatrix} -6 \\ -7 \end{bmatrix} \)
b. \( \begin{bmatrix} 14 \\ -3 \end{bmatrix} \)
c. \( \begin{bmatrix} 14 \\ 7 \end{bmatrix} \)
d. \( \begin{bmatrix} 15 \\ 6 \end{bmatrix} \)

Answer 1

To find the sum of the two vectors, we simply need to add the corresponding components of the vectors together.

So if we have vectors \( \mathbf{a} = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} \) and \( \mathbf{b} = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} \), their sum \( \mathbf{c} = \mathbf{a} + \mathbf{b} \) would have components \( c_1 = a_1 + b_1 \) and \( c_2 = a_2 + b_2 \). Specifically, for the given vectors: \( \mathbf{a} = \begin{bmatrix} 4 \\ -5 \end{bmatrix} \) and \( \mathbf{b} = \begin{bmatrix} 10 \\ 2 \end{bmatrix} \), we have \( \mathbf{a} + \mathbf{b} = \begin{bmatrix} 4 + 10 \\ -5 + 2 \end{bmatrix} = \begin{bmatrix} 14 \\ -3 \end{bmatrix} \).

Therefore, the sum of the vectors is: \( \mathbf{c} = \begin{bmatrix} 14 \\ -3 \end{bmatrix} \), which corresponds to option b.