## Linear Algebra: A Modern Introduction

If u and v are vectors, it follows: $\cos\theta=\frac{u \cdot v}{|u||v|}$ Therefore, we find: $u\cdot v=2(1)+(-1)(-2)+1(-1)=3$ $|u|=\sqrt{((2)^2+(-1)^2+1^2)}=\sqrt{6}$ $|v|=\sqrt{((1)^2+(-2)^2+(-1)^2)}=\sqrt{6}$ so $\cos\theta=\frac{u\cdot v}{|u||v|}=\frac{3}{\sqrt{6}\sqrt{6}}=\frac{3}{6}=\frac{1}{2}$ $\theta =60 \ degrees$ This is less than 90 degrees, so the angle is acute.