Question

For the vectors 𝑟⃗1 = 2𝑖̂− 3 𝑗̂+ 4𝑘̂ 𝑎𝑛𝑑 𝑟2̂ = 4𝑖̂+ 2𝑘̂

(a) Find the angle between them.

(b) Find the cross-product of the vectors.

Answer 1

(a) Find the dot product and magnitudes:

$$\vec r_1·\vec r_2=2·4+(-3)·2=2.\\ |\vec r_1|=\sqrt{2^2+(-3)^2}=3.6\\ |\vec r_2|=\sqrt{4^2+2^2}=4.5.$$

Find the angle between them:

$$\cos\theta=\frac{\vec r_1·\vec r_2}{|\vec r_1|·|\vec r_2|}=\frac{2}{3.6·4.5}=0.12,\\\space\\ \theta=83.1°$$

(b) Find the cross-product of the vectors:

$$\vec r_1\times\vec r_2=|\vec r_1|·|\vec r_2|\sin\theta=1.95$$