Question

1. let $\theta$ be the angle between vectors $\mathbf{u}$ and $\mathbf{v}$. given that $|\mathbf{u}| = 3, |\mathbf{v}| = 5$, and $(2\mathbf{u} - \mathbf{v}) \cdot (\mathbf{u} + \mathbf{v}) = 2$, find $\cos \theta$.

Explanation:

Step1: Expand the dot product

$(2\mathbf{u}-\mathbf{v})\cdot(\mathbf{u}+\mathbf{v}) = 2\mathbf{u}\cdot\mathbf{u} + 2\mathbf{u}\cdot\mathbf{v} - \mathbf{v}\cdot\mathbf{u} - \mathbf{v}\cdot\mathbf{v}$
Simplify using $\mathbf{u}\cdot\mathbf{v}=\mathbf{v}\cdot\mathbf{u}$:
$=2|\mathbf{u}|^2 + \mathbf{u}\cdot\mathbf{v} - |\mathbf{v}|^2$

Step2: Substitute known values

Given $|\mathbf{u}|=3$, $|\mathbf{v}|=5$, and the dot product equals 2:
$2 = 2(3)^2 + \mathbf{u}\cdot\mathbf{v} - (5)^2$
Calculate the squared magnitudes:
$2 = 2(9) + \mathbf{u}\cdot\mathbf{v} - 25$
$2 = 18 + \mathbf{u}\cdot\mathbf{v} - 25$

Step3: Solve for $\mathbf{u}\cdot\mathbf{v}$

Simplify the right-hand side:
$2 = \mathbf{u}\cdot\mathbf{v} - 7$
Rearrange to isolate $\mathbf{u}\cdot\mathbf{v}$:
$\mathbf{u}\cdot\mathbf{v} = 2 + 7 = 9$

Step4: Use dot product definition

Recall $\mathbf{u}\cdot\mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta$. Substitute known values:
$9 = (3)(5)\cos\theta$
$9 = 15\cos\theta$

Step5: Solve for $\cos\theta$

Rearrange to solve for $\cos\theta$:
$\cos\theta = \frac{9}{15} = \frac{3}{5}$

Answer:

$\frac{3}{5}$