To solve this problem, we need to find the value of [tex]\( |\hat{A} - \hat{B}| \)[/tex] given the equation [tex]\( 3|\vec{A} \times \vec{B}| = 4 \vec{A} \cdot \vec{B} \)[/tex].
First, let's express [tex]\( |\vec{A} \times \vec{B}| \)[/tex] and [tex]\( \vec{A} \cdot \vec{B} \)[/tex] in terms of magnitudes and the angle [tex]\(\theta\)[/tex] between vector [tex]\(\vec{A}\)[/tex] and vector [tex]\(\vec{B}\)[/tex].
1. Magnitudes and Angle Relation:
[tex]\[
|\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}|\sin(\theta)
\][/tex]
[tex]\[
\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos(\theta)
\][/tex]
2. Substitute into the Given Equation:
Given [tex]\( 3|\vec{A} \times \vec{B}| = 4 \vec{A} \cdot \vec{B} \)[/tex], we substitute the expressions:
[tex]\[
3|\vec{A}||\vec{B}|\sin(\theta) = 4|\vec{A}||\vec{B}|\cos(\theta)
\][/tex]
3. Simplify by Dividing Both Sides by [tex]\( |\vec{A}||\vec{B}| \)[/tex]:
[tex]\[
3\sin(\theta) = 4\cos(\theta)
\][/tex]
4. Isolate [tex]\(\tan(\theta)\)[/tex]:
[tex]\[
\frac{3\sin(\theta)}{3\cos(\theta)} = \frac{4\cos(\theta)}{3\cos(\theta)}
\][/tex]
[tex]\[
\tan(\theta) = \frac{4}{3}
\][/tex]
Thus, [tex]\(\theta\)[/tex] is the angle whose tangent is [tex]\(\frac{4}{3}\)[/tex].
5. Next, Calculate the Unit Vector Magnitude Difference:
We will find [tex]\( |\hat{A} - \hat{B}| \)[/tex]. Using the property of unit vectors:
[tex]\[
|\hat{A} - \hat{B}| = \sqrt{2 - 2\cos(\theta)}
\][/tex]
6. Find [tex]\(\cos(\theta)\)[/tex]:
Knowing [tex]\(\tan(\theta)\)[/tex], we calculate the cosine. Let:
[tex]\[
\theta = \tan^{-1}\left(\frac{4}{3}\right)
\][/tex]
Using [tex]\(\tan(\theta)\)[/tex], we have:
[tex]\[
\cos(\theta) = \frac{1}{\sqrt{1 + \tan^2(\theta)}} = \frac{1}{\sqrt{1 + \left(\frac{4}{3}\right)^2}} = \frac{1}{\sqrt{1 + \frac{16}{9}}} = \frac{1}{\sqrt{\frac{25}{9}}} = \frac{3}{5}
\][/tex]
7. Substitute [tex]\(\cos(\theta)\)[/tex] into the Magnitude Formula:
[tex]\[
|\hat{A} - \hat{B}| = \sqrt{2 - 2\cos(\theta)} = \sqrt{2 - 2 \cdot \frac{3}{5}} = \sqrt{2 - \frac{6}{5}} = \sqrt{\frac{10}{5} - \frac{6}{5}} = \sqrt{\frac{4}{5}}
\][/tex]
Hence, the value of [tex]\( |\hat{A} - \hat{B}| \)[/tex] is:
[tex]\(\boxed{\sqrt{\frac{4}{5}}}\)[/tex]
