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Section 11.2 : Vector Arithmetic

7. Prove the property : \(\vec v + \vec w = \vec w + \vec v\).

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These types of proofs always seem mysterious to students the first time they run across them. The main reason for the mystery is probably that it just seems obvious that it is true. That tends to make is difficult to prove.

We know that this property is true for numbers. However, we can’t assume that just because it’s true for numbers that it will be true for all other types of objects, vectors in this case!

So, let’s start off with two general vectors.

\[\vec v = \left\langle {{v_1},{v_2}, \ldots ,{v_n}} \right\rangle \hspace{0.75in}\vec w = \left\langle {{w_1},{w_2}, \ldots {w_n}} \right\rangle \]

To do this type of proof all we need to do is start with the left side perform the indicated operation, addition in this case, and then use properties about numbers that we already know to be true to try and manipulate it to look like the right side.

Show Step 2

So, let’s start off with the vector addition on the left side. All we want to do here is use the definition of vector addition to write the sum of the two vectors. This is,

\[\vec v + \vec w = \left\langle {{v_1},{v_2}, \ldots ,{v_n}} \right\rangle + \left\langle {{w_1},{w_2}, \ldots {w_n}} \right\rangle = \left\langle {{v_1} + {w_1},{v_2} + {w_2}, \ldots ,{v_n} + {w_n}} \right\rangle \] Show Step 3

Okay, as we noted above we know that \(2 + 3 = 3 + 2\). In other words, we know that the order we do addition of numbers doesn’t matter.

Why bring this up again?

Well, note that each of the components of the “new” vector on the right side is just a sum of two numbers. Therefore, we can use this property to flip the order of the addition in each of the components.

Doing this gives,

\[\vec v + \vec w = \left\langle {{v_1} + {w_1},{v_2} + {w_2}, \ldots ,{v_n} + {w_n}} \right\rangle = \left\langle {{w_1} + {v_1},{w_2} + {v_2}, \ldots ,{w_n} + {v_n}} \right\rangle \] Show Step 4

Now, recall that according to the definition of vector arithmetic the first number in the sum in each component of the vector on the right is the component of the first vector while the second number in the sum is the component of the second vector.

So, all we need to do now is “undo” the sum that gave the vector on the right to get,

\[\vec v + \vec w = \left\langle {{w_1} + {v_1},{w_2} + {v_2}, \ldots ,{w_n} + {v_n}} \right\rangle = \vec w + \vec v\]

This is exactly what we were asked to prove and so we are done!