Question Video: Identifying Algebraic Expressions for Vector Subtraction Physics • First Year of Secondary School

Video Transcript

If 𝐀 and 𝐁 are any two vectors, is the following algebraic expression true? 𝐀 minus 𝐁 equals negative 𝐁 minus 𝐀.

There are two ways we can go about verifying whether this expression is true graphically or algebraically. Let’s start by doing this graphically.

In order for this statement to be generally true, it must be true for any two vectors 𝐀 and 𝐁 that we could possibly choose. So, it doesn’t really matter what 𝐀 and 𝐁 are because it should be true no matter what. Let’s choose to define 𝐀 and 𝐁 like this. First, we’ll work out the resultant vector of the left-hand side of this expression, 𝐀 minus 𝐁. To do this, we need to subtract vector 𝐁 from vector 𝐀. We might recall that we can easily add vectors by lining them up tip to tail. But how do we subtract vectors?

Well, we can think about this expression another way. Just like with normal numbers, subtracting a vector is the same as adding its negative. So, 𝐀 minus 𝐁 can be rewritten as 𝐀 plus negative 𝐁. Negative 𝐁 is the vector that has the same magnitude as 𝐁 but that points in the opposite direction. So, we would draw negative 𝐁 like this. Now all we need to do is add negative 𝐁 to 𝐀 by positioning them tip to tail like so. The result is equal to this vector here, which we’ll call 𝐕 sub L to remind us it’s from the left-hand side of the equation.

Next, let’s look at the right-hand side of the equation, negative 𝐁 minus 𝐀. Let’s start by finding the vector inside the brackets, 𝐁 minus 𝐀. Just like before, we can rewrite this as 𝐁 plus negative 𝐀, and we can draw the vector negative 𝐀 like so. Now we add this to vector 𝐁, which leaves us with this resultant vector. However, we have to be careful not to forget the minus sign in front of the brackets. This tells us we need to reverse the direction of this vector like so. This gets us our final vector 𝐕 sub R.

Now, let’s compare our two vectors, 𝐕 L and 𝐕 R. We can see that they’re the same. In other words, we see that the resultant vector from the left-hand side of the equation is equal to the resultant vector from the right-hand side of the equation. Hence, we have verified that this expression is true for these two vectors that we chose. However, we could have approached this question in a different way that wouldn’t involve us drawing out so many vectors. If we approach this question algebraically, we can test whether this expression is true for any two vectors beyond just two random vectors that we think up.

Recall that a vector can be expressed algebraically in terms of its horizontal and vertical components, 𝑥 and 𝑦, and the horizontal and vertical unit vectors, 𝐢 hat and 𝐣 hat. We can rewrite vector 𝐀 as 𝐀 equals 𝐴 sub 𝑥 𝐢 hat plus 𝐴 sub 𝑦 𝐣 hat and vector 𝐁 as 𝐵 sub 𝑥 𝐢 hat plus 𝐵 sub 𝑦 𝐣 hat. Let’s sub these expressions into the left-hand side of the equation we want to verify. We find that 𝐀 minus 𝐁 is equal to 𝐴 sub 𝑥 𝐢 hat plus 𝐴 sub 𝑦 𝐣 hat minus 𝐵 sub 𝑥 𝐢 hat plus 𝐵 sub 𝑦 𝐣 hat, which equals 𝐴 sub 𝑥 𝐢 hat plus 𝐴 sub 𝑦 𝐣 hat minus 𝐵 sub 𝑥 𝐢 hat minus 𝐵 sub 𝑦 𝐣 hat. Now, let’s gather like terms and factor out the coefficients of 𝐢 hat and 𝐣 hat. This leaves us with 𝐴 sub 𝑥 minus 𝐵 sub 𝑥 𝐢 hat plus 𝐴 sub 𝑦 minus 𝐵 sub 𝑦 𝐣 hat.

Next, let’s think about the right-hand side of the expression, negative 𝐁 minus 𝐀. If we substitute in our expressions for 𝐀 and 𝐁, we find this is equal to the negative of 𝐵 sub 𝑥 𝐢 hat plus 𝐵 sub 𝑦 𝐣 hat minus 𝐴 sub 𝑥 𝐢 hat plus 𝐴 sub 𝑦 𝐣 hat. If we expand out all the brackets, paying attention to the two minus signs, we find this simplifies to negative 𝐵 sub 𝑥 𝐢 hat minus 𝐵 sub 𝑦 𝐣 hat plus 𝐴 sub 𝑥 𝐢 hat plus 𝐴 sub 𝑦 𝐣 hat. Again, if we gather like terms and factor out the coefficients of 𝐢 hat and 𝐣 hat, we’re left with 𝐴 sub 𝑥 minus 𝐵 sub 𝑥 𝐢 hat plus 𝐴 sub 𝑦 minus 𝐵 sub 𝑦 𝐣 hat as our expression for the right-hand side. This is exactly the same as we found for the left-hand side of the equation. So, we know the equation must be correct.

We have now verified the expression 𝐀 minus 𝐁 equals negative 𝐁 minus 𝐀 using two different methods. The answer to this question is therefore yes. This expression is true for any two vectors.