Video Transcript
Given that 𝐀 is the vector negative four, five and the vector 𝐀 plus the vector 𝐁 is the vector two, seven, find the vector 𝐁.
In this question, we’re given a vector 𝐀 and we’re told what happens when we add a vector 𝐁 to this vector. We need to use this information to determine the vector 𝐁. And there’s many different ways we can find this vector. For example, we can notice that we can add vector 𝐁 to vector 𝐀. And we can recall we can only add vectors together of the same dimension. So this means that vector 𝐁 must be a two-dimensional vector. We can call this the vector 𝑥, 𝑦. We can then use this to find an expression for vector 𝐀 plus vector 𝐁. It’s the vector negative four, five plus the vector 𝑥, 𝑦.
And now, we can recall that to add two vectors of the same dimension together, we just need to add the corresponding components together. Doing this gives us the vector negative four plus 𝑥, five plus 𝑦. And now, remember, in the question we’re told that vector 𝐀 plus vector 𝐁 is equal to two, seven. Therefore, we must have that the vector two, seven is equal to the vector negative four plus 𝑥, five plus 𝑦. And we can use this equation to find the values of 𝑥 and 𝑦. Remember, for two vectors to be equal, they need to have the same dimension and all of the corresponding components must be equal. And since both these vectors already have the same dimension, we just need to equate the components. We get two is equal to negative four plus 𝑥 and seven is equal to five plus 𝑦.
We can now solve each of these equations separately to determine the values of 𝑥 and 𝑦. Let’s start with the first equation. We add four to both sides of the equation to get that 𝑥 is equal to six. And in our second equation, we can subtract five from both sides of the equation to get that 𝑦 is equal to two. Therefore, we’ve shown 𝐁 is the vector six, two. And in fact, we can check our answer by finding the vector 𝐀 plus the vector 𝐁 and checking that it’s equal to the vector two, seven.
However, it’s worth noting this is not the only way we could’ve determined the vector 𝐁. Since we’re told in the question that 𝐀 plus 𝐁 is equal to the vector two, seven, we can subtract the vector 𝐀 from both sides of this equation to find an expression for vector 𝐁. We see that 𝐁 is the vector two, seven minus the vector 𝐀, and we’re given the vector 𝐀 in the question. Therefore, 𝐁 is equal to the vector two, seven minus the vector negative four, five. And we can now just evaluate the right-hand side of this equation.
Remember, to subtract two vectors of the same dimension, we just need to subtract the corresponding components. This gives us the vector two minus negative four, seven minus five. And if we evaluate the expression for each component, we get the vector six, two, which agrees with our other answer.
Therefore, we were able to show if 𝐀 is the vector negative four, five and the vector 𝐀 plus the vector 𝐁 is equal to the vector two, seven, then the vector 𝐁 must be equal to six, two.
