Answer to Question #128377 in Linear Algebra for Sym

Question

If a=i -j+2k, b = 4j - 2k and c = -10i - 2j +4k
Find the angle between the vectors a and c

Find |a +c |

Answer 1

$$\theta=\cos^{-1}\bigg(\frac{\vec{a}\cdot \vec{c}}{ac}\bigg)$$

Thus, $\vec a\cdot \vec c=-10+2+8=0\implies \theta=\cos^{-1}(0)=90^\circ$

$$\vec a +\vec c=(\hat i-\hat j+2\hat k)+(-10\hat i-2\hat j+4\hat k)=-9\hat i-3\hat j+6\hat k\\ \implies |\vec a +\vec c|=\sqrt{(-9)^2+(-3)^2+6^2}=\sqrt{126}$$