Question

If \((\vec a+\vec b)\cdot(\vec a-\vec b)=8\) and \(|\vec a|=8|\vec b|\), then find \(|\vec a|\) and \(|\vec b|\).

Hint:

Use (a+b).(a-b) = |a|^2-|b|^2 and substitute |a|=8|b|.

Answer 1

Step 1: Key Formula:
\((\vec a+\vec b)\cdot(\vec a-\vec b)=|\vec a|^{2}-|\vec b|^{2}\).

Step 2: Setting up equations:
Let \(|\vec b|=k\), so \(|\vec a|=8k\). Then \(|\vec a|^{2}-|\vec b|^{2}=64k^{2}-k^{2}=63k^{2}=8\).

Step 3: Solving for k:
\(k^{2}=\dfrac{8}{63}\ \Rightarrow\ k=\sqrt{\dfrac{8}{63}}=\dfrac{2\sqrt2}{3\sqrt7}=\dfrac{2\sqrt{14}}{21}\).

Final Answer:
\(|\vec b|=\dfrac{2\sqrt{14}}{21}\) and \(|\vec a|=8|\vec b|=\dfrac{16\sqrt{14}}{21}\).\[ \boxed{|\vec a|=\dfrac{16\sqrt{14}}{21},\ |\vec b|=\dfrac{2\sqrt{14}}{21}} \]