Question

In Exercises 20-23, find the dot product of the given pairs of vectors and the angle between the two vectors.

Short answer

The dot product is$-1$ and the angle isrole="math" localid="1649664356753" ${\cos }^{-1}\left(\frac{-1}{\sqrt{1711}}\right)$.

Step-by-step solution

Step 1. Given information.

The given pairs of vectors are:

Step 2. Find the dot product.

The dot product is: $\mathit{u}\mathbf{·}\mathit{v}=u_1{v}_1+u_2{v}_2+u_3{v}_3$

Therefore, the dot product of the given two vectors is$-1$.

Step 3. Find the angle between the two vectors.

The formula for the angle between the two vectors is:

$\cos \theta =\frac{\mathit{u}\mathbf{·}\mathit{v}}{||\mathit{u}||||\mathit{v}||}$

Then,

role="math" localid="1649664305708" $$\begin{gathered} \cos \theta =\frac{\mathit{u}\mathbf{·}\mathit{v}}{||\mathit{u}||||\mathit{v}||} \\ \cos \theta =\frac{-1}{\sqrt{29}\sqrt{59}} \\ \theta ={\cos }^{-1}\left(\frac{-1}{\sqrt{1711}}\right) \end{gathered}$$

Therefore, the angle isrole="math" localid="1649664318896" ${\cos }^{-1}\left(\frac{-1}{\sqrt{1711}}\right)$.