Question

how do i.... Find the angle between each pair of vectors below. (1 point each) a. a = 2x + 3y a = 4x + 2y

Answer 1

Recall that the dot product between two vectors $\mathbf a$ and $\mathbf b$ satisfies

$\mathbf a\cdot\mathbf b=\|\mathbf a\|\|\mathbf b\|\cos\theta$

where $\|\mathbf x\|$ denotes the norm of a vector $\mathbf x$, and $\theta$ is the angle between the two vectors.

So if $\mathbf a=(2,3)$ and $\mathbf b=(4,2)$, then

$(2,3)\cdot(4,2)=\|(2,3)\|\|(4,2)\|\cos\theta$

$\implies2\cdot4+3\cdot2=\sqrt{2^2+3^2}\sqrt{4^2+2^2}\cos\theta$

$\implies\cos\theta=\dfrac{14}{\sqrt{13}\sqrt{20}}$

$\implies\theta\approx29.7^\circ$