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: Expanding the dot product identity:
\((\vec a+\vec b)\cdot(\vec a-\vec b)=\vec a\cdot\vec a-\vec b\cdot\vec b=|\vec a|^2-|\vec b|^2=8\).

Step 2: Substituting the ratio directly:
With \(|\vec a|=8|\vec b|\), \((8|\vec b|)^2-|\vec b|^2=8\Rightarrow 63|\vec b|^2=8\).

Step 3: Isolating each magnitude:
\(|\vec b|=\sqrt{8/63}=\dfrac{2\sqrt{14}}{21}\), and multiplying by 8 gives \(|\vec a|=\dfrac{16\sqrt{14}}{21}\).

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