Question

find $mathbf{u} cdot mathbf{v}$ where $\theta$ is the angle between $mathbf{u}$ and $mathbf{v}$. round your answer to four decimals.
$\\|mathbf{u}\\|=16,\\|mathbf{v}\\|=5, \theta=\frac{pi}{4}$

Explanation:

Step1: Recall the formula for the dot product

The formula for the dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is \(\mathbf{u}\cdot\mathbf{v}=\|\mathbf{u}\|\|\mathbf{v}\|\cos\theta\).

Step2: Substitute the given values into the formula

Given \(\|\mathbf{u}\| = 16\), \(\|\mathbf{v}\| = 5\), and \(\theta=\frac{\pi}{4}\).
We know that \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\approx0.7071\).
Substitute into the formula: \(\mathbf{u}\cdot\mathbf{v}=16\times5\times\cos\frac{\pi}{4}\).

Step3: Calculate the value

\(\mathbf{u}\cdot\mathbf{v}=16\times5\times0.7071 = 80\times0.7071=56.5680\)

Answer:

\(56.5680\)