I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 11.3 : Dot Product
5. Determine the angle between \(\vec a = \vec i + 3\vec j - 2\vec k\) and \(\vec b = \left\langle { - 9,1, - 5} \right\rangle \).
Show SolutionNot really a whole lot to do here. All we really need to do is rewrite the formula from the geometric interpretation of the dot product as,
\[\cos \theta = \frac{{\vec a\centerdot \vec b}}{{\left\| {\vec a} \right\|\,\,\left\| {\vec b} \right\|}}\]This will allow us to quickly determine the angle between the two vectors.
We’ll first need the following quantities (we’ll leave it to you to verify the arithmetic involved in these computations….).
\[\vec a\centerdot \vec b = 4\hspace{0.25in}\hspace{0.25in}\left\| {\vec a} \right\| = \sqrt {14} \hspace{0.25in}\hspace{0.25in}\left\| {\vec b} \right\| = \sqrt {107} \]The angle between the vectors is then,
\[\cos \theta = \frac{4}{{\sqrt {14} \,\sqrt {107} }} = 0.1033\hspace{0.25in} \Rightarrow \hspace{0.25in}\theta = {\cos ^{ - 1}}\left( {0.1034} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{1.4673\,\,{\rm{radians}}}}\]