Question

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

Trigonometry
ANSWERED
asked 2021-08-16
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?

Expert Answers (1)

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
\(\displaystyle\theta={128.1}^{\circ}\)
26
 
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