Question

If $\overrightarrow a = 5i - j - 3k$ and $\overrightarrow b = i + 3j - 5k$, then show that vectors $\overrightarrow a + \overrightarrow b$ and $\overrightarrow a - \overrightarrow b$ are perpendicular.

Answer 1

In the given question, we have been given that there are two vectors. We have to prove that the two vectors representing the sum of the two vectors and the difference of the two vectors are perpendicular. We are going to solve it by using the defining formula of two vectors to be perpendicular.

Complete step by step answer:
The given vectors are $\overrightarrow a = 5i - j - 3k$ and $\overrightarrow b = i + 3j - 5k$.
We have to prove that,
$\overrightarrow a + \overrightarrow b = \overrightarrow m = \left( {5i - j - 3k} \right) + \left( {i + 3j - 5k} \right) = 6i + 2j - 8k$ and
$\overrightarrow a - \overrightarrow b = \overrightarrow n = \left( {5i - j - 3k} \right) - \left( {i + 3j - 5k} \right) = 4i - 4j + 2k$ are perpendicular.
Now, two vectors are perpendicular if their scalar product is zero.
So, we are going to find the scalar product of $\overrightarrow m$ and $\overrightarrow n$.
Now, $\overrightarrow m .\overrightarrow n = \left( {6i + 2j - 8k} \right).\left( {4i - 4j + 2k} \right) = 24 - 8 - 16 = 0$
Hence, $\overrightarrow a + \overrightarrow b$ and $\overrightarrow a - \overrightarrow b$ are perpendicular.

Note:
In the given question, we were given two vectors. There was no condition or any special thing whatsoever about the two given vectors. We had to prove that two vectors which were represented by the sum of the original two vectors and the difference of the original two vectors were perpendicular. We solved it by first writing the sum and the difference equal to any variable which was in turn a reference for this system. Now, two vectors are perpendicular if their scalar product is zero. So, we found their scalar product. And it got out to be zero, so the two vectors were perpendicular.