Answer to Question #265721 in Physics for Bex

Question

Two vectors are given: 𝑥⃗ = 3𝑖⃗ + 2𝑗⃗, 𝑦⃗ = 2𝑖⃗ − 8𝑗⃗.

a) Draw these vectors in Cartesian coordinate system.

b) Represent both of these vectors in magnitude, direction notation.

c) Represent vector 2𝑥⃗ + 3𝑦⃗ in magnitude, direction notation.

Answer 1

a) The diagram:

b) The magnitude-direction rotation:

$$|x|=\sqrt{3^2+2^2}=3.6,\\\space\\ \theta_x=\arctan\frac{2}{3}=34°,\\\space\\ |y|=\sqrt{2^2+(-8)^2}=8.2,\\\space\\ \theta_y=\arctan\frac{-8}{2}=-76°.$$

c) The magnitude-direction rotation:

$$|2x+3y|=\sqrt{(2·3+3·2)^2+(2·2+3(-8))^2}=23,\\\space\\ \theta_{x+y}=\arctan\frac{2·3+3·2}{2·3+3·2}=-59°.$$