Question

given vectors ( u = (-3, 2) ) and ( v = (2, 1) ), what is the measure of the angle between the vectors?

( 7.1^{circ} )
( 60.3^{circ} )
( 119.7^{circ} )
( 172.9^{circ} )

Explanation:

Step1: Calculate the dot - product

The dot - product formula is \(\mathbf{u}\cdot\mathbf{v}=u_1v_1 + u_2v_2\).
For \(\mathbf{u}=(-3,2)\) and \(\mathbf{v}=(2,1)\), \(\mathbf{u}\cdot\mathbf{v}=(-3)\times2+2\times1=-6 + 2=-4\).

Step2: Calculate the magnitudes

The magnitude formula is \(|\mathbf{a}|=\sqrt{a_1^{2}+a_2^{2}}\).
\(|\mathbf{u}|=\sqrt{(-3)^{2}+2^{2}}=\sqrt{9 + 4}=\sqrt{13}\), \(|\mathbf{v}|=\sqrt{2^{2}+1^{2}}=\sqrt{4+1}=\sqrt{5}\).

Step3: Use the formula for the angle between two vectors

The formula is \(\cos\theta=\frac{\mathbf{u}\cdot\mathbf{v}}{|\mathbf{u}||\mathbf{v}|}\).
Substitute the values: \(\cos\theta=\frac{-4}{\sqrt{13}\times\sqrt{5}}=\frac{-4}{\sqrt{65}}\approx - 0.496\).

Step4: Find the angle

\(\theta=\cos^{-1}(-0.496)\approx119.7^{\circ}\) (using a calculator to find the inverse - cosine value).

Answer:

C. \(119.7^{\circ}\)