# Question Video: Properties of Operations on Vectors Mathematics

Complete the following: 2(〈2, 5〉 + 〈5, 1〉) = 〈＿, ＿〉 + 〈10, 2〉.

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### 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.