If vector a = 5i - j - 3k and vector b = i + 3j - 5k, then show that the vectors vector a + vector b and vector a - vector b are orthogonal.

Question

If  \(\vec a=5\hat i-\hat j-3\hat k\)  and  \(\vec b = \hat i+3\hat j-5\hat k\),  then show that the vectors  \(\vec a+\vec b\) and  \(\vec a-\vec b\) are orthogonal.

Answer 1

Meaning of orthogonal is that two vectors are perpendicular to each other, so their dot product is zero.

So, to satisfy the orthogonal condition  \(\vec a.\vec b=0\)

Hence proved

Answer 2

a = 5i - j - 3k

b = i + 3j - 5k

a + b = 6i +2j - 8k

a - b = 4i - 4j + 2k

(a+b)(a-b) = 24 - 8 - 16 = 0

As dot product = 0, angle between them = 90

Hence proved

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