Question

If $(\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = 27$ and $|\vec{a}| = 2|\vec{b}|$, then $|\vec{b}|$ is:

• A. 3
• B. 2
• C. $\frac{5}{6}$
• D. 6

Answer 1

Answer: (A)

3

Answer 2

We are given that $|\vec{a}| = 2|\vec{b}|$. To find $|\vec{a} - \vec{b}|$, we use the following approach:
First, recall the formula for the magnitude of the difference between two vectors:
$|\vec{a} - \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 - 2\vec{a} \cdot \vec{b}}.$
Since $|\vec{a}| = 2|\vec{b}|$, we have:
$|\vec{a}|^2 = 4|\vec{b}|^2.$
Now, let’s compute the dot product $\vec{a} \cdot \vec{b}$. The vectors $\vec{a}$ and $\vec{b}$ are in the same direction, so $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| = 2|\vec{b}|^2$.
Now substitute these values into the magnitude formula:
$|\vec{a} - \vec{b}| = \sqrt{4|\vec{b}|^2 + |\vec{b}|^2 - 2 \cdot 2|\vec{b}|^2}.$
Simplifying:
$|\vec{a} - \vec{b}| = \sqrt{4|\vec{b}|^2 + |\vec{b}|^2 - 4|\vec{b}|^2} = \sqrt{|\vec{b}|^2} = |\vec{b}|.$
Since $|\vec{b}| = 3$, we get:
$|\vec{a} - \vec{b}| = 3.$
Thus, the correct answer is: 3.