Question Video: Adding Vectors to Solve Vector Equations Mathematics

Video Transcript

Given that the vector 𝐀 is seven, negative one and the vector 𝐀 minus the vector 𝐁 is three, negative two, find the vector 𝐀 plus the vector 𝐁.

In this question, we’re given the vector 𝐀 and we’re given the vector 𝐀 minus the vector 𝐁. We need to use this information to determine the sum of the vectors 𝐀 and 𝐁. There’s a few different ways of going about answering this question; we’ll only go through one of these. We can see that we need to find the vector 𝐀 plus the vector 𝐁 and we’re given the vector 𝐀. So we can do this by finding an expression for the vector 𝐁. This means we’ll need to use the two pieces of information we’re given to determine the vector 𝐁.

Let’s start with the piece of information we’re given about vector 𝐁: 𝐀 minus 𝐁 is equal to the vector three, negative two. We want to use this information to determine the vector 𝐁. And since this is an equation, we can do this by rearranging our equation. We’ll start by adding the vector 𝐁 to both sides of the equation. On the left-hand side of our equation, we get the vector 𝐀 minus the vector 𝐁 plus the vector 𝐁. And remember, a vector 𝐁 minus the vector 𝐁 will be equal to the zero vector. So the left-hand side of our equation simplifies to give us 𝐀 plus the zero vector. But adding the zero vector doesn’t change the value of our vector. So the left-hand side of this equation is just the vector 𝐀. And this will be equal to the vector three, negative two plus the vector 𝐁.

Next, to make the vector 𝐁 the subject of this equation, we need to subtract the vector three, negative two from both sides of the equation. Doing this, the left-hand side of this equation becomes 𝐀 minus the vector three, negative two. And the right-hand side of our equation becomes the vector three minus negative two minus the vector three minus negative two plus the vector 𝐁. And we know subtracting a vector from itself gives us the zero vector. So the right-hand side of this equation simplifies to give us the vector 𝐁.

Now, we can find the vector 𝐁 by recalling that we’re given that the vector 𝐀 is the vector seven, negative one. Substituting this into our equation, we get the vector 𝐁 is equal to the vector seven, negative one minus the vector three, negative two. And now, we’re ready to find the vector 𝐁. We recall to subtract two vectors of the same dimension, we just need to subtract the corresponding components. This gives us that 𝐁 is the vector seven minus three, negative one minus negative two. And now we just evaluate the expression for each of our components. We get that 𝐁 is the vector four, one.

Now that we have expressions for vectors 𝐀 and 𝐁, we can just add the two vectors together to find the vector 𝐀 plus the vector 𝐁. We get that 𝐀 plus 𝐁 is equal to the vector seven, negative one plus the vector four, one. And we recall to add two vectors of the same dimension together, we just need to add the corresponding components together. This gives us the vector seven plus four, negative one plus one. And finally, we just evaluate the expression for each of the components. We get the vector 11, zero.

Therefore, we were able to show if 𝐀 is the vector seven, negative one and the vector 𝐀 minus the vector 𝐁 is equal to the vector three, negative two, then 𝐀 plus 𝐁 must be equal to the vector 11, zero.