Question

Find the angle between the given vectors. Round your answer, in degrees, to two decimal places.u = (-7,2), v = (-3, -4)

Answer 1

In order to calculate the angle between the vectors, we need to use the dot product:

`u\cdot v=|u||v|\cos\theta=u_xv_x+u_yv_y`

Calculating the magnitude of each vector, we have:

`\begin{gathered} |u|=√(u_x^2+u_y^2)=√(49+4)=√(53)\\ \\ |v|=√(v_x^2+v_y^2)=√(9+16)=√(25)=5 \end{gathered}`

Now, calculating the dot product, we have:

`u\cdot v=u_xv_x+u_yv_y=(-7)\cdot(-3)+2\cdot(-4)=21-8=13`

So, calculating the cosine of theta and then the angle theta, we have:

`\begin{gathered} \cos\theta=(u\cdot v)/(|u||v|)=(13)/(5√(53))=(13)/(36.4)=0.35714\\ \\ \theta=\cos^(-1)(0.35714)\\ \\ \theta=69.07° \end{gathered}`