If `vec(a)= hati - hatj, bar(b) = hat(i) + hat(j), vec(c) = hat(i) + 3hat(j) + 5hat(k)` and `vec(n)` be a unit vector such that `vec(b).vec(n) =0, vec(a).vec(n) = 0` then value of `|vec(c).vec(n)|` is

Options

A. 1
B. 3
C. 5
D. 2

Answer

To solve the given problem step by step, we will follow the outlined approach based on the information provided in the question.

### Step 1: Define the vectors
We have the following vectors:
- \( \vec{a} = \hat{i} - \hat{j} \)
- \( \vec{b} = \hat{i} + \hat{j} \)
- \( \vec{c} = \hat{i} + 3\hat{j} + 5\hat{k} \)

### Step 2: Understand the conditions
We know that:
- \( \vec{b} \cdot \vec{n} = 0 \) (which means \( \vec{n} \) is perpendicular to \( \vec{b} \))
- \( \vec{a} \cdot \vec{n} = 0 \) (which means \( \vec{n} \) is also perpendicular to \( \vec{a} \))
- \( \vec{n} \) is a unit vector, so \( |\vec{n}| = 1 \)

### Step 3: Find the cross product \( \vec{a} \times \vec{b} \)
To find a vector \( \vec{n} \) that is perpendicular to both \( \vec{a} \) and \( \vec{b} \), we can calculate the cross product \( \vec{a} \times \vec{b} \).

Using the determinant method:
\[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -1 & 0 \\ 1 & 1 & 0 \end{vmatrix} \]

Calculating this determinant:
- The \( \hat{i} \) component: \( 0 - 0 = 0 \)
- The \( \hat{j} \) component: \( 0 - 0 = 0 \)
- The \( \hat{k} \) component: \( 1 \cdot 1 - (-1) \cdot 1 = 1 + 1 = 2 \)

Thus,
\[ \vec{a} \times \vec{b} = 2\hat{k} \]

### Step 4: Express \( \vec{n} \)
Since \( \vec{n} \) is a unit vector and is in the direction of \( \vec{a} \times \vec{b} \), we can express it as:
\[ \vec{n} = \lambda (2\hat{k}) \]
where \( \lambda \) is a scalar to ensure \( |\vec{n}| = 1 \).

### Step 5: Find \( \lambda \)
To find \( \lambda \), we set the magnitude of \( \vec{n} \) to 1:
\[ |\vec{n}| = |2\lambda \hat{k}| = 2|\lambda| = 1 \implies |\lambda| = \frac{1}{2} \]
Thus, we can take \( \lambda = \frac{1}{2} \), leading to:
\[ \vec{n} = \hat{k} \]

### Step 6: Calculate \( |\vec{c} \cdot \vec{n}| \)
Now we need to find \( |\vec{c} \cdot \vec{n}| \):
\[ \vec{c} \cdot \vec{n} = (\hat{i} + 3\hat{j} + 5\hat{k}) \cdot \hat{k} \]
Calculating the dot product:
\[ \vec{c} \cdot \vec{n} = 0 + 0 + 5 = 5 \]

### Step 7: Final answer
Thus, the value of \( |\vec{c} \cdot \vec{n}| \) is:
\[ |\vec{c} \cdot \vec{n}| = 5 \]