Sure, let's work through this problem step by step.
Given vectors are:
[tex]\[
\overrightarrow{A} = 2 \hat{\imath} + 3 \hat{\jmath} + \hat{k}
\][/tex]
[tex]\[
\overrightarrow{B} = 2 \hat{\imath} - 3 \hat{\jmath} + \hat{k}
\][/tex]
First, we need to find the difference between the two vectors [tex]\(\overrightarrow{A} - \overrightarrow{B}\)[/tex].
Perform the subtraction component-wise:
[tex]\[
(2 \hat{\imath} + 3 \hat{\jmath} + \hat{k}) - (2 \hat{\imath} - 3 \hat{\jmath} + \hat{k})
\][/tex]
Subtracting the corresponding components:
[tex]\[
(2 - 2) \hat{\imath} + (3 - (-3)) \hat{\jmath} + (1 - 1) \hat{k}
\][/tex]
Simplifying this:
[tex]\[
0 \hat{\imath} + 6 \hat{\jmath} + 0 \hat{k}
\][/tex]
So, the difference vector is:
[tex]\[
\overrightarrow{A} - \overrightarrow{B} = 0 \hat{\imath} + 6 \hat{\jmath} + 0 \hat{k} = 6 \hat{\jmath}
\][/tex]
Next, we need to find the magnitude of this difference vector [tex]\(\overrightarrow{D} = 0 \hat{\imath} + 6 \hat{\jmath} + 0 \hat{k}\)[/tex].
The magnitude of a vector is calculated using the formula:
[tex]\[
\| \overrightarrow{D} \| = \sqrt{(D_x)^2 + (D_y)^2 + (D_z)^2}
\][/tex]
For the vector [tex]\(\overrightarrow{D} = 0 \hat{\imath} + 6 \hat{\jmath} + 0 \hat{k}\)[/tex], we have:
[tex]\[
\| \overrightarrow{D} \| = \sqrt{(0)^2 + (6)^2 + (0)^2} = \sqrt{0 + 36 + 0} = \sqrt{36} = 6
\][/tex]
Therefore, the magnitude of [tex]\(\overrightarrow{A} - \overrightarrow{B}\)[/tex] is:
[tex]\[
6
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{6}
\][/tex]
