Question

Find the angle between the vectors \(\vec a=\hat i+\hat j-\hat k\) and \(\vec b=\hat i-\hat j+\hat k\).

Hint:

Use cos(theta) = (a.b)/(|a||b|).

Answer 1

Step 1: Direct component computation:
\(\vec a\cdot \vec b=1(1)+1(-1)+(-1)(1)=-1\), and each vector has magnitude \(\sqrt{1^2+1^2+(-1)^2}=\sqrt3\).

Step 2: Forming and simplifying the cosine ratio:
\(\cos\theta=\dfrac{-1}{\sqrt3\cdot\sqrt3}=\dfrac{-1}{3}\).

Final Answer:
\[ \boxed{\theta=\cos^{-1}\left(-\dfrac13\right)} \]