Find `'lambda'` when the vectors : `vec(a)=2hat(i)+lambda hat(j)+hat(k)` and `vec(b)=hat(i)-2hat(j)+3hat(k)` are perpendicular to each other.

Answer

To find the value of \( \lambda \) when the vectors \( \vec{a} = 2\hat{i} + \lambda \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} \) are perpendicular to each other, we can follow these steps:

### Step 1: Understand the condition for perpendicularity
Two vectors are perpendicular if their dot product is zero. Therefore, we need to calculate the dot product \( \vec{a} \cdot \vec{b} \) and set it equal to zero.

### Step 2: Write the dot product formula
The dot product of two vectors \( \vec{a} = x_1 \hat{i} + x_2 \hat{j} + x_3 \hat{k} \) and \( \vec{b} = y_1 \hat{i} + y_2 \hat{j} + y_3 \hat{k} \) is given by:
\[ \vec{a} \cdot \vec{b} = x_1 y_1 + x_2 y_2 + x_3 y_3 \]
For our vectors:
- \( x_1 = 2, x_2 = \lambda, x_3 = 1 \)
- \( y_1 = 1, y_2 = -2, y_3 = 3 \)

### Step 3: Substitute the values into the dot product
Now, substituting the values into the dot product formula:
\[ \vec{a} \cdot \vec{b} = (2)(1) + (\lambda)(-2) + (1)(3) \]
This simplifies to:
\[ \vec{a} \cdot \vec{b} = 2 - 2\lambda + 3 \]

### Step 4: Set the dot product equal to zero
Since the vectors are perpendicular, we set the dot product equal to zero:
\[ 2 - 2\lambda + 3 = 0 \]

### Step 5: Simplify the equation
Combine the constants:
\[ 5 - 2\lambda = 0 \]

### Step 6: Solve for \( \lambda \)
Rearranging the equation gives:
\[ -2\lambda = -5 \]
Dividing both sides by -2:
\[ \lambda = \frac{5}{2} \]

### Final Answer
The value of \( \lambda \) is \( \frac{5}{2} \).

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