Question

# Given v=-7i-6j and w=5i-j, find the angle between v and w. What

Trigonometry
Given $$\displaystyle{v}=-{7}{i}-{6}{j}$$ and $$\displaystyle{w}={5}{i}-{j}$$, find the angle between v and w.
What is the angle between v and w?

2021-08-17
Given that $$\displaystyle{v}=-{7}{i}-{6}{j}$$ and $$\displaystyle{w}={5}{i}-{j}$$
As we know that
$$\displaystyle{v}\cdot{w}={\left|{v}\right|}{\left|{w}\right|}{\cos{\theta}}$$
$$\displaystyle{\cos{\theta}}={\frac{{{v}\cdot{w}}}{{{\left|{v}\right|}{\left|{w}\right|}}}}$$
$$\displaystyle{\cos{\theta}}={\frac{{{\left(-{7}{i}-{6}{j}\right)}{\left({5}{i}-{j}\right)}}}{{{\left|-{7}{i}-{6}{j}\right|}{\left|{5}{i}-{j}\right|}}}}$$
$$\displaystyle{\cos{\theta}}={\frac{{{\left(-{7}\times{5}\right)}+{\left(-{6}\times-{1}\right)}}}{{\sqrt{{{\left(-{7}\right)}^{{2}}+{\left(-{6}\right)}^{{2}}}}\cdot\sqrt{{{5}^{{2}}+{\left(-{1}\right)}^{{2}}}}}}}$$
$$\displaystyle{\cos{\theta}}={\frac{{-{35}+{6}}}{{\sqrt{{{85}}}\cdot\sqrt{{{26}}}}}}$$
$$\displaystyle{\cos{\theta}}={\frac{{-{29}}}{{\sqrt{{{85}}}\cdot\sqrt{{{26}}}}}}$$
$$\displaystyle{\cos{\theta}}=-{0.61688}$$
Take cos inverse
$$\displaystyle\theta={{\cos}^{{-{1}}}{\left(-{0.61688}\right)}}$$
$$\displaystyle\theta={{\cos}^{{-{1}}}{\left(-{0.61688}\right)}}$$
Round nearest tenth
$$\displaystyle\theta={128.1}^{\circ}$$