Question

given vectors ( u = langle -5, 1
angle ) and ( v = langle 8, 6
angle ), which statement is true for ( u ) and ( v )?
the vectors point in the same direction.
the vectors form an acute angle.
the vectors form an obtuse angle.
the vectors point in opposite directions.

Explanation:

Step1: Calculate the dot - product

The dot - product formula is \(u\cdot v = u_1v_1+u_2v_2\).
For \(u=\langle - 5,1
angle\) and \(v = \langle8,6
angle\), we have \(u\cdot v=(-5)\times8 + 1\times6=-40 + 6=-34\).

Step2: Calculate the magnitudes

The magnitude formula is \(\vert u\vert=\sqrt{u_1^{2}+u_2^{2}}\) and \(\vert v\vert=\sqrt{v_1^{2}+v_2^{2}}\).
\(\vert u\vert=\sqrt{(-5)^{2}+1^{2}}=\sqrt{25 + 1}=\sqrt{26}\), \(\vert v\vert=\sqrt{8^{2}+6^{2}}=\sqrt{64 + 36}=\sqrt{100} = 10\).

Step3: Use the formula for the cosine of the angle between two vectors

The formula is \(\cos\theta=\frac{u\cdot v}{\vert u\vert\vert v\vert}\).
Substitute the values: \(\cos\theta=\frac{-34}{\sqrt{26}\times10}\approx\frac{-34}{50.99}\approx - 0.67\).
Since \(-1<\cos\theta<0\), the angle \(\theta\) (where \(0\leq\theta\leq\pi\)) is obtuse.

Answer:

C. The vectors form an obtuse angle.