Let `vecu= hati + hatj , vecv = hati -hatja and hati -hatj and vecw =hati + 2hatj + 3 hatk` If ` hatn` isa unit vector such that `vecu .hatn=0 and vecn .hatn =0 , " then " |vecw.hatn|` is equal to

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the given problem step by step, we will follow the mathematical operations involving vectors.

### Given:
- \(\vec{u} = \hat{i} + \hat{j}\)
- \(\vec{v} = \hat{i} - \hat{j}\)
- \(\vec{w} = \hat{i} + 2\hat{j} + 3\hat{k}\)

### Step 1: Find the cross product \(\vec{u} \times \vec{v}\)
To find the unit vector \(\hat{n}\) that is perpendicular to both \(\vec{u}\) and \(\vec{v}\), we first calculate the cross product \(\vec{u} \times \vec{v}\).

\[ \vec{u} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 0 \\ 1 & -1 & 0 \end{vmatrix} \]

Calculating the determinant:

\[ = \hat{i}(1 \cdot 0 - 0 \cdot (-1)) - \hat{j}(1 \cdot 0 - 0 \cdot 1) + \hat{k}(1 \cdot (-1) - 1 \cdot 1) \]

\[ = \hat{i}(0) - \hat{j}(0) + \hat{k}(-1 - 1) \]

\[ = -2\hat{k} \]

### Step 2: Find the unit vector \(\hat{n}\)
Now, we can express the unit vector \(\hat{n}\) as:

\[ \hat{n} = \frac{\vec{u} \times \vec{v}}{|\vec{u} \times \vec{v}|} \]

Calculating the magnitude of \(\vec{u} \times \vec{v}\):

\[ |\vec{u} \times \vec{v}| = |-2\hat{k}| = 2 \]

Thus, the unit vector \(\hat{n}\) is:

\[ \hat{n} = \frac{-2\hat{k}}{2} = -\hat{k} \]

### Step 3: Calculate \(\vec{w} \cdot \hat{n}\)
Now we need to find \(|\vec{w} \cdot \hat{n}|\):

\[ \vec{w} \cdot \hat{n} = (\hat{i} + 2\hat{j} + 3\hat{k}) \cdot (-\hat{k}) \]

Calculating the dot product:

\[ = \hat{i} \cdot (-\hat{k}) + 2\hat{j} \cdot (-\hat{k}) + 3\hat{k} \cdot (-\hat{k}) \]

\[ = 0 + 0 - 3 = -3 \]

### Step 4: Find the magnitude
The magnitude is:

\[ |\vec{w} \cdot \hat{n}| = |-3| = 3 \]

### Final Answer
Thus, \(|\vec{w} \cdot \hat{n}| = 3\).

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