Question

what is the relationship between vectors u = (1, 0) and v = (0, -3)? the vectors form an acute angle of approximately 48°. the vectors form an acute angle of approximately 71°. the vectors are orthogonal because the angle between them is 90°. the vectors form an obtuse angle of approximately 132°.

Explanation:

Step1: Calculate the dot - product

The dot - product formula for two vectors \(\mathbf{u}=(u_1,u_2)\) and \(\mathbf{v}=(v_1,v_2)\) is \(\mathbf{u}\cdot\mathbf{v}=u_1v_1 + u_2v_2\).
For \(\mathbf{u}=(1,0)\) and \(\mathbf{v}=(0, - 3)\), we have \(\mathbf{u}\cdot\mathbf{v}=(1)\times(0)+(0)\times(-3)=0\).

Step2: Use the dot - product property

The formula for the angle \(\theta\) between two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is \(\cos\theta=\frac{\mathbf{u}\cdot\mathbf{v}}{\vert\mathbf{u}\vert\vert\mathbf{v}\vert}\).
Since \(\mathbf{u}\cdot\mathbf{v} = 0\), then \(\cos\theta=\frac{0}{\vert\mathbf{u}\vert\vert\mathbf{v}\vert}=0\).
The magnitude of \(\mathbf{u}\) is \(\vert\mathbf{u}\vert=\sqrt{1^{2}+0^{2}} = 1\), and the magnitude of \(\mathbf{v}\) is \(\vert\mathbf{v}\vert=\sqrt{0^{2}+(-3)^{2}}=3\).
We know that if \(\cos\theta = 0\), then \(\theta = 90^{\circ}\). Two vectors with an angle of \(90^{\circ}\) between them are orthogonal.

Answer:

The vectors are orthogonal because the angle between them is \(90^{\circ}\).