Given vectors $\mathbf{a}$ and $\mathbf{b}$ such that $\|\mathbf{a}\| = 6,$ $\|\mathbf{b}\| = 8,$ and $\|\mathbf{a} + \mathbf{b}\| = 11.$  Find $\cos \theta,$ where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}.$
We have that
\begin{align*}
\|\mathbf{a} + \mathbf{b}\|^2 &= (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b}) \\
&= \mathbf{a} \cdot \mathbf{a} + 2 \mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{b} \\
&= \|\mathbf{a}\|^2 + 2 \mathbf{a} \cdot \mathbf{b} + \|\mathbf{b}\|^2.
\end{align*}Hence, $11^2 = 6^2 + 2 \mathbf{a} \cdot \mathbf{b} + 8^2,$ so
\[\mathbf{a} \cdot \mathbf{b} = \frac{21}{2}.\]Then
\[\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|} = \frac{21/2}{6 \cdot 8} = \boxed{\frac{7}{32}}.\]
\frac{7}{32}
Level 3
Precalculus
Problem: Given vectors $\mathbf{a}$ and $\mathbf{b}$ such that $\|\mathbf{a}\| = 6,$ $\|\mathbf{b}\| = 8,$ and $\|\mathbf{a} + \mathbf{b}\| = 11.$  Find $\cos \theta,$ where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}.$
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find $\cos \theta$, we start with the formula for the magnitude of the sum of two vectors: 

\[
\|\mathbf{a} + \mathbf{b}\|^2 = \|\mathbf{a}\|^2 + \|\mathbf{b}\|^2 + 2\|\mathbf{a}\|\|\mathbf{b}\|\cos \theta.
\]

Substituting the known magnitudes into the equation, we have:

\[
\|\mathbf{a} + \mathbf{b}\|^2 = 11^2 = 121,
\]
\[
\|\mathbf{a}\|^2 = 6^2 = 36,
\]
\[
\|\mathbf{b}\|^2 = 8^2 = 64.
\]

Now, substituting these values into the equation gives:

\[
121 = 36 + 64 + 2 \cdot 6 \cdot 8 \cdot \cos \theta.
\]

Calculating the sum of the squares:

\[
121 = 100 + 96 \cos \theta.
\]

Rearranging the equation to isolate the cosine term:

\[
121 - 100 = 96 \cos \theta,
\]
\[
21 = 96 \cos \theta.
\]

Now, solving for $\cos \theta$:

\[
\cos \theta = \frac{21}{96}.
\]

To simplify this fraction, we find the greatest common divisor of 21 and 96, which is 3:

\[
\cos \theta = \frac{7}{32}.
\]

Thus, the final answer is $\frac{7}{32}$.<|END_OF_LATENT|> The final answer is $\frac{7}{32}$.