What is the value of b such that the scalar product of the vector `hat(i) + hat(j) + hat(k)` with the unit vector parallel to the sum of the vectors `2hat(i) + 4hat(j)-5hat(k) and b hat(i) + 2hat(j) + 3hat(k)` is unity ?

Options

A. `-2`
B. `-1`
C. 0
D. 1

Answer

To solve the problem, we need to find the value of \( b \) such that the scalar product of the vector \( \hat{i} + \hat{j} + \hat{k} \) with the unit vector parallel to the sum of the vectors \( 2\hat{i} + 4\hat{j} - 5\hat{k} \) and \( b\hat{i} + 2\hat{j} + 3\hat{k} \) equals 1.

### Step 1: Find the sum of the vectors
First, we need to add the two vectors:
\[ \text{Vector 1} = 2\hat{i} + 4\hat{j} - 5\hat{k} \]
\[ \text{Vector 2} = b\hat{i} + 2\hat{j} + 3\hat{k} \]
The sum of these vectors is:
\[ (2 + b)\hat{i} + (4 + 2)\hat{j} + (-5 + 3)\hat{k} = (2 + b)\hat{i} + 6\hat{j} - 2\hat{k} \]

### Step 2: Find the magnitude of the resultant vector
Next, we need to calculate the magnitude of the resultant vector:
\[ \text{Magnitude} = \sqrt{(2 + b)^2 + 6^2 + (-2)^2} \]
Calculating each term:
\[ = \sqrt{(2 + b)^2 + 36 + 4} = \sqrt{(2 + b)^2 + 40} \]

### Step 3: Find the unit vector
The unit vector in the direction of the resultant vector is given by:
\[ \text{Unit Vector} = \frac{(2 + b)\hat{i} + 6\hat{j} - 2\hat{k}}{\sqrt{(2 + b)^2 + 40}} \]

### Step 4: Set up the scalar product equation
We need to find the scalar product of \( \hat{i} + \hat{j} + \hat{k} \) with the unit vector and set it equal to 1:
\[ (\hat{i} + \hat{j} + \hat{k}) \cdot \left( \frac{(2 + b)\hat{i} + 6\hat{j} - 2\hat{k}}{\sqrt{(2 + b)^2 + 40}} \right) = 1 \]
Calculating the dot product:
\[ \frac{(2 + b) + 6 - 2}{\sqrt{(2 + b)^2 + 40}} = 1 \]
This simplifies to:
\[ \frac{(b + 6)}{\sqrt{(2 + b)^2 + 40}} = 1 \]

### Step 5: Solve for \( b \)
Now, squaring both sides:
\[ (b + 6)^2 = (2 + b)^2 + 40 \]
Expanding both sides:
\[ b^2 + 12b + 36 = b^2 + 4b + 4 + 40 \]
This simplifies to:
\[ b^2 + 12b + 36 = b^2 + 4b + 44 \]
Cancelling \( b^2 \) from both sides:
\[ 12b + 36 = 4b + 44 \]
Rearranging gives:
\[ 12b - 4b = 44 - 36 \]
\[ 8b = 8 \]
Thus, we find:
\[ b = 1 \]

### Final Answer
The value of \( b \) is \( 1 \).