Question

I was hoping you could help me with this question?

Answer 1

In order to determine the angle between the given vectors, use the following formula:

`\vec{u}\cdot\vec{v}=uv\cos \theta`

where the left hand side of the equation is the dot product of u and v vectors, u and v are the magnitude of the vectors and θ is the angle between the vectors.

Then, by solving for θ, you obtain:

`\begin{gathered} \cos \theta=\frac{\vec{u}\cdot\vec{v}}{uv} \\ \theta=\cos ^(-1)(\frac{\vec{u}\cdot\vec{v}}{uv}) \end{gathered}`

Then, first calculate the dot product:

`\begin{gathered} \vec{u}\cdot\vec{v}=(u_x)(v_x)+(u_y)(v_y) \\ \vec{u}\cdot\vec{v}=(0)(11)+(-10)(-12)=120 \end{gathered}`

The magnitudes of the vectors are:

`\begin{gathered} u=\sqrt[]{u^2_x+u^2_y}=\sqrt[]{(0)^{}+(-10)^2}=\sqrt[]{100}=10.00 \\ v=\sqrt[]{v^2_x+v^2_y}=\sqrt[]{(11)^2+(-12)^2}=\sqrt[]{265}\approx16.28 \end{gathered}`

Then, by replacing the dot product and the valued of u and v into the expression for θ, you obtain:

`\theta=\cos ^(-1)((120)/(10.00\cdot16.28))\approx42.51`

Hence, the angle between u and v vectors is approximately 42.51 degrees.