### Instruction: Find $\begin{pmatrix} 1 \\ 4 \\ -6 \end{pmatrix} + \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}.$.### Response: This problem is asking me to add two vectors in $\mathbb{R}^3$, the space of ordered triples of real numbers.
To add two vectors, I can simply add their corresponding components, or entries, and write the result as another vector with the same shape.
So, I start with the first component of both vectors: 1 and 2.
I add them together and get 3.
That will be the first component of the sum vector.
Next, I move to the second component of both vectors: 4 and -1.
I add them together and get 3.
That will be the second component of the sum vector.
Finally, I look at the third component of both vectors: -6 and 3.
I add them together and get -3.
That will be the third component of the sum vector.
I write the sum vector as an ordered triple in parentheses, or as a column vector in brackets.
Either way, the answer is the same.