Let `vec(U)=hati+hatj,vecV=hati-hatjand vec(W)=3hati+5hatj+3hatk. ` If `hat(n)` is a unit vector such that `vec(U)`.`hat(n)` =0 and `vec(V)`.`hat(n)`=0 , then `|vecW.hatn|` is equal to

Answer

To solve the problem, we need to follow these steps:

### Step 1: Identify the vectors
We are given the vectors:
- \(\vec{U} = \hat{i} + \hat{j}\)
- \(\vec{V} = \hat{i} - \hat{j}\)
- \(\vec{W} = 3\hat{i} + 5\hat{j} + 3\hat{k}\)

### Step 2: Find the unit vector \(\hat{n}\)
The unit vector \(\hat{n}\) is perpendicular to both \(\vec{U}\) and \(\vec{V}\). Therefore, we can find \(\hat{n}\) by calculating the cross product \(\vec{U} \times \vec{V}\).

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

### Step 3: Calculate the determinant
Calculating the determinant gives us:

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

\[ = 0\hat{i} - 0\hat{j} - 2\hat{k} = -2\hat{k} \]

### Step 4: Normalize the vector to find \(\hat{n}\)
Since \(\hat{n}\) is a unit vector, we normalize \(-2\hat{k}\):

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

### Step 5: Calculate \(|\vec{W} \cdot \hat{n}|\)
Now, we need to find \(|\vec{W} \cdot \hat{n}|\):

\[ \vec{W} \cdot \hat{n} = (3\hat{i} + 5\hat{j} + 3\hat{k}) \cdot (-\hat{k}) \]

Calculating the dot product:

\[ = 3(\hat{i} \cdot -\hat{k}) + 5(\hat{j} \cdot -\hat{k}) + 3(\hat{k} \cdot -\hat{k}) \]

Since \(\hat{i} \cdot \hat{k} = 0\) and \(\hat{j} \cdot \hat{k} = 0\):

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

### Step 6: Find the magnitude
Finally, we take the absolute value:

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

### Final Answer
Thus, the value of \(|\vec{W} \cdot \hat{n}|\) is **3**.

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