Let `vec(u) = hat(i) + hat(j), vec(v) = hat(i) - hat(j)` and `vec(w) = hat(i) + 2hat(j) + 3hat(k)` . If `hat(n)` is a unit vector such that `vec(u).vec(n)` = 0 and `vec(v).hat(n) = 0` then `|vec(w).hat(n)|` 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 outlined process to find the value of \(|\vec{w} \cdot \hat{n}|\).

### Step 1: Define the vectors
We have the following vectors:
- \(\vec{u} = \hat{i} + \hat{j}\)
- \(\vec{v} = \hat{i} - \hat{j}\)
- \(\vec{w} = \hat{i} + 2\hat{j} + 3\hat{k}\)

### Step 2: Define the unit vector \(\hat{n}\)
Let \(\hat{n} = x\hat{i} + y\hat{j} + z\hat{k}\). Since \(\hat{n}\) is a unit vector, it satisfies the condition:
\[ x^2 + y^2 + z^2 = 1 \]

### Step 3: Use the condition \(\vec{u} \cdot \hat{n} = 0\)
Calculating the dot product:
\[ \vec{u} \cdot \hat{n} = (\hat{i} + \hat{j}) \cdot (x\hat{i} + y\hat{j} + z\hat{k}) = x + y + 0 = 0 \]
This gives us our first equation:
\[ x + y = 0 \quad \text{(1)} \]

### Step 4: Use the condition \(\vec{v} \cdot \hat{n} = 0\)
Calculating the dot product:
\[ \vec{v} \cdot \hat{n} = (\hat{i} - \hat{j}) \cdot (x\hat{i} + y\hat{j} + z\hat{k}) = x - y + 0 = 0 \]
This gives us our second equation:
\[ x - y = 0 \quad \text{(2)} \]

### Step 5: Solve the equations
From equations (1) and (2):
1. From (1): \(y = -x\)
2. From (2): \(y = x\)

Setting these equal gives:
\[ -x = x \implies 2x = 0 \implies x = 0 \]
Substituting \(x = 0\) into equation (1):
\[ 0 + y = 0 \implies y = 0 \]

### Step 6: Find \(z\)
Now substituting \(x\) and \(y\) into the unit vector condition:
\[ 0^2 + 0^2 + z^2 = 1 \implies z^2 = 1 \implies z = 1 \quad \text{(since \(z\) is positive for a unit vector)} \]
Thus, we have:
\[ \hat{n} = 0\hat{i} + 0\hat{j} + 1\hat{k} = \hat{k} \]

### Step 7: Calculate \(|\vec{w} \cdot \hat{n}|\)
Now we need to find \(|\vec{w} \cdot \hat{n}|\):
\[ \vec{w} \cdot \hat{n} = (\hat{i} + 2\hat{j} + 3\hat{k}) \cdot \hat{k} = 0 + 0 + 3 = 3 \]

Thus, the final answer is:
\[ |\vec{w} \cdot \hat{n}| = 3 \]

### Final Answer
\[ |\vec{w} \cdot \hat{n}| = 3 \]