### Video Transcript

Complete the following: the vector
one, nine plus the vector five, two is equal to the vector five, two plus what.

We begin here by simplifying the
left-hand side of our equation. We begin by recalling that to find
the sum of a pair of vectors, we simply add their corresponding components. In this question, to add the
vectors one, nine and five, two, we add one and five and then separately nine and
two. This means that the left-hand side
of our equation is equal to the vector six, 11. If we let the missing vector on the
right-hand side have components 𝑥 and 𝑦, we can simplify the right-hand side as
shown. The vector five, two plus the
vector 𝑥, 𝑦 gives us the vector five plus 𝑥, two plus 𝑦.

We can now equate the two sides of
our equation. Six, 11 is equal to five plus 𝑥,
two plus 𝑦. For the two vectors to be equal, we
know that their corresponding components must be equal. This gives us two equations we need
to solve: six is equal to five plus 𝑥 and 11 is equal to two plus 𝑦. Subtracting five from both sides of
our first equation, we see that 𝑥 is equal to one. And subtracting two from both sides
of our second equation, we see that 𝑦 is equal to nine. The missing vector is therefore
equal to one, nine. The vector one, nine plus the
vector five, two is equal to the vector five, two plus the vector one, nine.