Question

find a · b if a = <-4,3,-2> and b = <-5,3,-1>
a · b =
is the angle between the vectors \acute\, \obtuse\ or
ight\?

Explanation:

Step1: Recall dot - product formula

For $\mathbf{a}=\langle a_1,a_2,a_3
angle$ and $\mathbf{b}=\langle b_1,b_2,b_3
angle$, $\mathbf{a}\cdot\mathbf{b}=a_1b_1 + a_2b_2+a_3b_3$.
Given $\mathbf{a}=\langle - 4,3,-2
angle$ and $\mathbf{b}=\langle - 5,3,-1
angle$, we have $\mathbf{a}\cdot\mathbf{b}=(-4)\times(-5)+3\times3+(-2)\times(-1)$.

Step2: Calculate the dot - product

$\mathbf{a}\cdot\mathbf{b}=20 + 9+2=31$.

Step3: Determine the angle type

We know that $\mathbf{a}\cdot\mathbf{b}=\vert\mathbf{a}\vert\vert\mathbf{b}\vert\cos\theta$, where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$. Since $\mathbf{a}\cdot\mathbf{b}=31>0$, and $\vert\mathbf{a}\vert>0$, $\vert\mathbf{b}\vert>0$, then $\cos\theta>0$. When $\cos\theta>0$, $0\leq\theta<\frac{\pi}{2}$, so the angle is acute.

Answer:

1. 31
2. acute