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Step-by-Step Solution
Step 1: Understand the Problem
We are given three vectors:
$\overrightarrow{u} = \hat{i} + \hat{j}$
$\overrightarrow{v} = \hat{i} - \hat{j}$
$\overrightarrow{w} = \hat{i} + 2\hat{j} + 3\hat{k}$
We need to find $|\overrightarrow{w} \cdot \hat{n}|$ where $\hat{n}$ is a unit vector that is perpendicular to both $\overrightarrow{u}$ and $\overrightarrow{v}$ (i.e., $\overrightarrow{u} \cdot \hat{n} = 0$ and $\overrightarrow{v} \cdot \hat{n} = 0$).
Step 2: Express the Unit Vector $\hat{n}$
Since $\hat{n}$ is perpendicular to both $\overrightarrow{u}$ and $\overrightarrow{v}$, it must be in the direction of their cross product. Hence,
$\displaystyle \hat{n} \;=\; \frac{\overrightarrow{u} \times \overrightarrow{v}}{\bigl|\overrightarrow{u} \times \overrightarrow{v}\bigr|}.
$
Step 3: Use the Scalar Triple Product
We want $|\overrightarrow{w} \cdot \hat{n}|$. Using properties of dot and cross products, we can write:
$\displaystyle
|\overrightarrow{w} \cdot \hat{n}|
=
\left|\overrightarrow{w} \cdot \bigl( \frac{\overrightarrow{u} \times \overrightarrow{v}}{\lvert \overrightarrow{u} \times \overrightarrow{v} \rvert} \bigr)\right|
=
\frac{\bigl|\overrightarrow{w} \cdot (\overrightarrow{u} \times \overrightarrow{v})\bigr|}{\bigl|\overrightarrow{u} \times \overrightarrow{v}\bigr|}.
$
Thus, we need to compute:
The scalar triple product $\overrightarrow{w} \cdot (\overrightarrow{u} \times \overrightarrow{v})$.
The magnitude $\bigl|\overrightarrow{u} \times \overrightarrow{v}\bigr|$.
Step 4: Compute $\overrightarrow{u} \times \overrightarrow{v}$
$\displaystyle
\overrightarrow{u} = \langle 1, \, 1, \, 0 \rangle, \quad
\overrightarrow{v} = \langle 1, \,-1, \, 0 \rangle.
$
The cross product is given by the determinant:
$\displaystyle
\overrightarrow{u} \times \overrightarrow{v}
=
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k}\\
1 & 1 & 0\\
1 & -1 & 0
\end{vmatrix}.
$
Expanding along the third column (noting that the $k$-components in $u$ and $v$ are zero) or along the standard formula for 3D cross product:
$\displaystyle
\overrightarrow{u} \times \overrightarrow{v}
= \bigl(1 \cdot 0 - 1 \cdot 0 \bigr)\hat{i}
- \bigl(1 \cdot 0 - 0 \cdot 1 \bigr)\hat{j}
+ \bigl(1 \cdot (-1) - 1 \cdot 1 \bigr)\hat{k}.
$
$\displaystyle
\overrightarrow{u} \times \overrightarrow{v}
= 0\,\hat{i} - 0\,\hat{j} + ( -1 - 1 ) \,\hat{k}
= -2 \hat{k}.
$
The magnitude is:
$\displaystyle
\bigl|\overrightarrow{u} \times \overrightarrow{v}\bigr|
= \sqrt{(-2)^2}
= 2.
$
Step 5: Compute $\overrightarrow{w} \cdot (\overrightarrow{u} \times \overrightarrow{v})$
$\displaystyle
\overrightarrow{w} = \langle 1,\, 2,\, 3\rangle, \quad
\overrightarrow{u} \times \overrightarrow{v} = \langle 0, \,0, \,-2\rangle.
$
$\displaystyle
\overrightarrow{w} \cdot (\overrightarrow{u} \times \overrightarrow{v})
= \langle 1,\, 2,\, 3\rangle
\cdot
\langle 0, \,0, \,-2\rangle
= (1 \times 0) + (2 \times 0) + (3 \times (-2))
= -6.
$
$\displaystyle
\bigl|\overrightarrow{w} \cdot (\overrightarrow{u} \times \overrightarrow{v})\bigr|
= 6.
$
Step 6: Combine the Results
Finally, we find:
$\displaystyle
|\overrightarrow{w} \cdot \hat{n}|
= \frac{ \bigl|\overrightarrow{w} \cdot (\overrightarrow{u} \times \overrightarrow{v})\bigr| }
{ \bigl|\overrightarrow{u} \times \overrightarrow{v}\bigr| }
= \frac{6}{2}
= 3.
$
Conclusion
The value of $|\overrightarrow{w} \cdot \hat{n}|$ is $\boxed{3}$.
