Question

question suppose that $vec{u}=langle - 4,3
angle$ and $vec{v}=langle - 4,-7
angle$. what is the angle between $vec{u}$ and $vec{v}$, rounded to the nearest degree? select the correct answer below: 97° 89° 95° 103°

Explanation:

Step1: Recall dot - product formula

The dot - product formula is $\vec{u}\cdot\vec{v}=\vert\vec{u}\vert\vert\vec{v}\vert\cos\theta$, where $\theta$ is the angle between $\vec{u}$ and $\vec{v}$. First, calculate the dot - product of $\vec{u}=(-4,3)$ and $\vec{v}=(-4,-7)$. $\vec{u}\cdot\vec{v}=(-4)\times(-4)+3\times(-7)=16 - 21=-5$.

Step2: Calculate magnitudes of vectors

The magnitude of a vector $\vec{a}=(x,y)$ is $\vert\vec{a}\vert=\sqrt{x^{2}+y^{2}}$. For $\vec{u}=(-4,3)$, $\vert\vec{u}\vert=\sqrt{(-4)^{2}+3^{2}}=\sqrt{16 + 9}=\sqrt{25}=5$. For $\vec{v}=(-4,-7)$, $\vert\vec{v}\vert=\sqrt{(-4)^{2}+(-7)^{2}}=\sqrt{16 + 49}=\sqrt{65}$.

Step3: Solve for $\cos\theta$

From $\vec{u}\cdot\vec{v}=\vert\vec{u}\vert\vert\vec{v}\vert\cos\theta$, we have $\cos\theta=\frac{\vec{u}\cdot\vec{v}}{\vert\vec{u}\vert\vert\vec{v}\vert}$. Substitute the values: $\cos\theta=\frac{-5}{5\times\sqrt{65}}=-\frac{1}{\sqrt{65}}\approx - 0.124$.

Step4: Find the angle $\theta$

$\theta=\arccos(-0.124)$. Using a calculator, $\theta\approx97^{\circ}$.

Answer:

A. $97^{\circ}$