1. **State the problem:** Given two vectors $\mathbf{v}_1 = \begin{pmatrix}5 \\ -1 \\ 3\end{pmatrix}$ and $\mathbf{v}_2 = \begin{pmatrix}1 \\ 1 \\ -5\end{pmatrix}$, we want to analyze their properties such as addition, scalar multiplication, or dot product. 2. **Formula and rules:** For vectors $\mathbf{a} = \begin{pmatrix}a_1 \\ a_2 \\ a_3\end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix}b_1 \\ b_2 \\ b_3\end{pmatrix}$: - Vector addition: $\mathbf{a} + \mathbf{b} = \begin{pmatrix}a_1 + b_1 \\ a_2 + b_2 \\ a_3 + b_3\end{pmatrix}$ - Scalar multiplication: $c \mathbf{a} = \begin{pmatrix}c a_1 \\ c a_2 \\ c a_3\end{pmatrix}$ - Dot product: $\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$ 3. **Calculate the dot product:** $$\mathbf{v}_1 \cdot \mathbf{v}_2 = 5 \times 1 + (-1) \times 1 + 3 \times (-5) = 5 - 1 - 15 = -11$$ 4. **Calculate the vector sum:** $$\mathbf{v}_1 + \mathbf{v}_2 = \begin{pmatrix}5 + 1 \\ -1 + 1 \\ 3 + (-5)\end{pmatrix} = \begin{pmatrix}6 \\ 0 \\ -2\end{pmatrix}$$ 5. **Interpretation:** - The dot product $-11$ indicates the vectors are neither orthogonal nor parallel. - The sum vector $\begin{pmatrix}6 \\ 0 \\ -2\end{pmatrix}$ is the resultant vector when $\mathbf{v}_1$ and $\mathbf{v}_2$ are added component-wise. **Final answer:** - Dot product: $-11$ - Sum vector: $\begin{pmatrix}6 \\ 0 \\ -2\end{pmatrix}$