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: Key Approach:
Two vectors are perpendicular if their dot product is \(0\). We use the identity \((\vec a+\vec b)\cdot(\vec a-\vec b)=|\vec a|^{2}-|\vec b|^{2}\).

Step 2: Computing the magnitudes:
\(|\vec a|^{2}=5^{2}+(-1)^{2}+(-3)^{2}=25+1+9=35\). \(|\vec b|^{2}=1^{2}+3^{2}+(-5)^{2}=1+9+25=35\).

Final Answer:
\((\vec a+\vec b)\cdot(\vec a-\vec b)=35-35=0\). Since the dot product is zero, \((\vec a+\vec b)\) and \((\vec a-\vec b)\) are perpendicular.\[ \boxed{(\vec a+\vec b)\cdot(\vec a-\vec b)=0} \]