Question

consider vectors u and v, where u = (1, -1) and v = (1, 1). what is the measure of the angle between the vectors? 0° 60° 90° 180°

Explanation:

Step1: Calculate the dot - product

The dot - product formula is \(u\cdot v = u_1v_1+u_2v_2\).
For \(u=(1, - 1)\) and \(v=(1,1)\), we have \(u\cdot v=(1\times1)+(-1\times1)=1 - 1=0\).

Step2: Use the formula for the angle between two vectors

The formula for the angle \(\theta\) between two vectors \(u\) and \(v\) is \(\cos\theta=\frac{u\cdot v}{\vert u\vert\vert v\vert}\).
Since \(u\cdot v = 0\), then \(\cos\theta=\frac{0}{\vert u\vert\vert v\vert}=0\).

Step3: Find the angle

We know that if \(\cos\theta = 0\) and \(0^{\circ}\leq\theta\leq180^{\circ}\), then \(\theta = 90^{\circ}\) (because \(y = \cos x\) and \(x\in[0,\pi]\), when \(y = 0\), \(x=\frac{\pi}{2}=90^{\circ}\)).

Answer:

\(90^{\circ}\)