Question

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

Hint:

Use (a+b).(a-b) = |a|^2 - |b|^2 and check if it is zero.

Answer 1

Step 1: Direct expansion:
\((\vec a+\vec b)\cdot(\vec a-\vec b)=\vec a\cdot\vec a-\vec a\cdot\vec b+\vec b\cdot\vec a-\vec b\cdot\vec b=|\vec a|^2-|\vec b|^2\) (the cross terms cancel since dot product is commutative).

Step 2: Substituting the magnitudes found directly from components:
\(|\vec a|^2=25+1+9=35\) and \(|\vec b|^2=1+9+25=35\), which are equal.

Final Answer:
So the dot product is \(35-35=0\), proving perpendicularity.\[ \boxed{\text{Perpendicular, since the dot product} = 0} \]