Question

find a · b if |a| = 14, |b| = 1, and the angle between a and b is π/3.

Explanation:

Step1: Recall dot - product formula

The dot - product of two vectors $\mathbf{a}$ and $\mathbf{b}$ is given by $\mathbf{a}\cdot\mathbf{b}=\vert\mathbf{a}\vert\vert\mathbf{b}\vert\cos\theta$, where $\vert\mathbf{a}\vert$ and $\vert\mathbf{b}\vert$ are the magnitudes of vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, and $\theta$ is the angle between them.

Step2: Substitute given values

We are given that $\vert\mathbf{a}\vert = 14$, $\vert\mathbf{b}\vert=1$, and $\theta=\frac{\pi}{3}$. Substituting these values into the formula $\mathbf{a}\cdot\mathbf{b}=\vert\mathbf{a}\vert\vert\mathbf{b}\vert\cos\theta$, we get $\mathbf{a}\cdot\mathbf{b}=14\times1\times\cos\frac{\pi}{3}$.

Step3: Evaluate the cosine value

Since $\cos\frac{\pi}{3}=\frac{1}{2}$, then $\mathbf{a}\cdot\mathbf{b}=14\times1\times\frac{1}{2}$.

Step4: Calculate the result

$14\times1\times\frac{1}{2}=7$.

Answer:

$7$