### Video Transcript

Complete the following: two
multiplied by the vector two, five plus the vector five, one is equal to what plus
the vector 10, two.

We begin this question by
simplifying the left-hand side of the equation. Firstly, we use the fact that
scalar multiplication is distributive over vector addition. This means that the left-hand side
becomes two multiplied by the vector two, five plus two multiplied by the vector
five, one. We can then evaluate the scalar
multiplication. Two multiplied by the vector two,
five is equal to the vector two multiplied by two, two multiplied by five. This is equal to four, 10. Likewise, multiplying the vector
five, one by the scalar two gives us the vector 10, two. We can then equate this to the
right-hand side of our equation and let the components of the unknown vector be 𝑥
and 𝑦.

Next, we can consider the
elimination property of vector addition, which states that if vector 𝐮 plus vector
𝐯 is equal to vector 𝐮 plus vector 𝐰, then vector 𝐯 is equal to vector 𝐰. The vector 10, two appears in the
sum on both sides of our equation. This means that the other vectors
on each side must also be equal. The vector four, 10 is equal to the
vector 𝑥, 𝑦. We can therefore conclude that the
missing vector is four, 10.