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: Key Formula:
\(\cos\theta=\dfrac{\vec a\cdot\vec b}{|\vec a||\vec b|}\).

Step 2: Computing the dot product and magnitudes:
\(\vec a\cdot\vec b=(1)(1)+(1)(-1)+(-1)(1)=1-1-1=-1\). \(|\vec a|=|\vec b|=\sqrt{1+1+1}=\sqrt3\).

Final Answer:
\(\cos\theta=\dfrac{-1}{3}\), so \(\theta=\cos^{-1}\!\Big(-\dfrac13\Big)\).\[ \boxed{\theta=\cos^{-1}\left(-\dfrac13\right)} \]