Suppose that 𝐀 is the vector four, seven, negative seven; 𝐁 is the vector negative five, one, negative two; and 𝐀 plus 𝐁 plus 𝐂 equals 𝐢 hat. What is 𝐂?
To answer this question, we need to understand the notation used to represent the vectors 𝐀, 𝐁, and 𝐢 hat, as well as how to compute the sum of three vectors. We can represent any vector 𝐕 as a list of its components inside of angular brackets. And this is the notation used to represent vectors 𝐀 and 𝐁. Equivalently, we can express the vector in terms of the three mutually perpendicular unit vectors 𝐢 hat, 𝐣 hat, and 𝐤 hat by multiplying each unit vector by the corresponding vector component.
In fact, as we’re using it, the bracket notation is simply a shorthand for the expansion in terms of unit vectors. And this expansion is the notation that we’re using when we represent 𝐢 hat on the right-hand side of our equation where we have 𝑉 one equals one and 𝑉 two and 𝑉 three are both zero. So one, zero, zero in the bracket notation would be exactly equivalent to 𝐢 hat in the unit vector notation. Similarly, zero, one, zero is 𝐣 hat, and zero, zero, one would be 𝐤 hat.
Anyway, when we add vectors, we always add component-wise. That is, we collect all of the 𝐢 hat components in a single term, all of the 𝐣 hat components in another term, and all of the 𝐤 hat components in a third term. Equivalently, we collect all of the components in each of the separate positions in the bracket into their own separate terms. Just for practice, let’s perform the sum using brackets but give our final answer using unit vectors.
We don’t know what 𝐂 is, so we’ll represent its components with the variables 𝐶 one, 𝐶 two, and 𝐶 three. Okay, let’s now carry out the addition. Remember, we do the calculation for each component on its own. For the first component, we take the first component in each of the brackets: four for 𝐀, negative five for 𝐁, and 𝐶 one for 𝐂. For the second component, we take the second components, which are seven, one, and 𝐶 two. Likewise, for the third component, we take the third components of each of the brackets, which are negative seven, negative two, and 𝐶 three.
Let’s simplify the parentheses a little bit. Four plus negative five is negative one. Seven plus one is eight. And negative seven plus negative two is negative nine. So the components of 𝐀 plus 𝐁 plus 𝐂 are negative one plus 𝐶 one, eight plus 𝐶 two, and negative nine plus 𝐶 three. And we know that this sum is equal to 𝐢 hat, which in brackets is represented as one, zero, zero. We now need to recall that, similar to the way that addition and subtraction are done component by component, when two vectors are equal to each other, the equality must hold component by component for both vectors.
So in order for 𝐀 plus 𝐁 plus 𝐂 to equal 𝐢 hat, it must be that the first component negative one plus 𝐶 one is equal to the first component of 𝐢 hat, which is one; the second component eight plus 𝐶 two is equal to the second component of 𝐢 hat, which is zero; and the third component negative nine plus 𝐶 three equals the third component of 𝐢 hat, which is also zero. We can now solve each of these three equations to find the three components of the vector 𝐂.
In the first equation, we add one to both sides to find that 𝐶 one is equal to two. In the second equation, we subtract eight from both sides to find that 𝐶 two is negative eight. And finally, for the last equation, adding nine to both sides gives us 𝐶 three equals nine. To convert these three components back into a vector, we either plug them into a bracket like we had expressed before or, what we’re going to do, use them as coefficients in an expansion in terms of unit vectors.
Matching each component to the corresponding unit vector, we find that 𝐂 is two 𝐢 hat minus eight 𝐣 hat plus nine 𝐤 hat.