Question Video: Adding and Subtracting Vectors to Solve Vector Equations Mathematics

Video Transcript

Given that 𝐀 is the vector three, negative two, 𝐁 is the vector negative five, four, and the vector 𝐀 minus the vector 𝐁 plus the vector 𝐂 is the vector six, negative one, find the vector 𝐂.

In this question, we’re given two vectors. We’re given the vector 𝐀, and we’re given the vector 𝐁. We want to determine the vector 𝐂, and we’re given an expression involving the vector 𝐂. We’re told 𝐀 minus 𝐁 plus 𝐂 is equal to six, negative one. Therefore, we can determine the vector 𝐂 by rearranging our equation to make the vector 𝐂 the subject. To do this, we first need to subtract the vector 𝐀 from both sides of the equation.

Well, it’s worth recalling if we subtract any vector from itself, we will get the zero vector. And adding and subtracting the zero vector from any other vector won’t change its value. Therefore, when we subtract 𝐀 from the left-hand side of our equation, we’ll just be left with negative 𝐁 plus 𝐂. And on the right-hand side of our equation, we’ll have the vector six negative one minus the vector 𝐀. But remember, we’re told in the question that 𝐀 is the vector three, negative two. Therefore, we have negative 𝐁 plus 𝐂 is equal to the vector six, negative one minus the vector three, negative two.

We can continue this process by adding the vector 𝐁 to both sides of the equation. This time, on the left-hand side of our equation, we’ll have the vector 𝐁 minus the vector 𝐁. And remember, a vector minus itself is just equal to the zero vector. And adding the zero vector to vector 𝐂 won’t change its value. Therefore, the left-hand side of this equation is just the vector 𝐂. We then add the vector 𝐁 to the right-hand side of this equation. Well, we remember we’re told in the question that 𝐁 is the vector negative five, four. Therefore, the vector 𝐂 is equal to the vector six, negative one minus the vector three, negative two plus the vector negative five, four.

And now, we can find the vector 𝐂. We recall to subtract two vectors of the same dimension, we just need to subtract the corresponding components. This would allow us to subtract the first two vectors. However, we can simplify this process by remembering to add two vectors together, we just need to add the corresponding components. We can actually do the vector subtraction and the vector addition in one step.

In the first component of this vector, we need to subtract three and add negative five. The first component is six minus three plus negative five. And in the second component, we need to subtract negative two and add four. The second component of this vector is negative one minus negative two plus four. Now, we can find the vector 𝐂 by evaluating both of these expressions. We get that 𝐂 is the vector negative two, five which is our final answer. Therefore, we were able to show if 𝐀 is the vector three, negative two; 𝐁 is the vector negative five, four; and the vector 𝐀 minus the vector 𝐁 plus the vector 𝐂 is the vector six, negative one, then 𝐂 is the vector negative two, five.