Question

find the angle between 2i - 5j and j.
the angle between 2i - 5j and j is 158.2°.
(round to the nearest tenth as needed.)

Explanation:

Step1: Recall the formula for the angle between two vectors

The formula for the angle \(\theta\) between two vectors \(\vec{u} = u_1\vec{i}+u_2\vec{j}\) and \(\vec{v}=v_1\vec{i} + v_2\vec{j}\) is \(\cos\theta=\frac{\vec{u}\cdot\vec{v}}{\vert\vec{u}\vert\vert\vec{v}\vert}\), where \(\vec{u}\cdot\vec{v}=u_1v_1 + u_2v_2\), \(\vert\vec{u}\vert=\sqrt{u_1^{2}+u_2^{2}}\) and \(\vert\vec{v}\vert=\sqrt{v_1^{2}+v_2^{2}}\).

Let \(\vec{u} = 2\vec{i}- 5\vec{j}\) and \(\vec{v}=\vec{j}=0\vec{i}+1\vec{j}\).

First, calculate the dot product \(\vec{u}\cdot\vec{v}\):
\(\vec{u}\cdot\vec{v}=(2)(0)+(- 5)(1)=- 5\)

Step2: Calculate the magnitudes of the vectors

Calculate \(\vert\vec{u}\vert\):
\(\vert\vec{u}\vert=\sqrt{2^{2}+(-5)^{2}}=\sqrt{4 + 25}=\sqrt{29}\approx5.385\)

Calculate \(\vert\vec{v}\vert\):
\(\vert\vec{v}\vert=\sqrt{0^{2}+1^{2}} = 1\)

Step3: Find the cosine of the angle

Substitute the values into the cosine formula:
\(\cos\theta=\frac{-5}{\sqrt{29}\times1}\approx\frac{-5}{5.385}\approx - 0.928\)

Step4: Find the angle

To find \(\theta\), we take the arccosine of \(- 0.928\):
\(\theta=\arccos(-0.928)\)
Using a calculator, \(\theta\approx158.2^{\circ}\) (rounded to the nearest tenth)

Answer:

\(158.2^{\circ}\)