The vectors `2hati+hatj-4hatk` and `ahati+bhatj+chatk` are perpendicular, if

Options

A. a=2, b=3, c=-4
B. a=4, b=4, c=5
C. a=4, b=4, c=-5
D. a=-4, b=4, c=-5

Answer

To determine the conditions under which the vectors \( \mathbf{u} = 2\hat{i} + \hat{j} - 4\hat{k} \) and \( \mathbf{v} = a\hat{i} + b\hat{j} + c\hat{k} \) are perpendicular, 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 of vectors \( \mathbf{u} \) and \( \mathbf{v} \) and set it to zero.

### Step 2: Calculate the dot product
The dot product \( \mathbf{u} \cdot \mathbf{v} \) is given by:
\[ \mathbf{u} \cdot \mathbf{v} = (2\hat{i} + \hat{j} - 4\hat{k}) \cdot (a\hat{i} + b\hat{j} + c\hat{k}) \]
Using the properties of dot products:
\[ \mathbf{u} \cdot \mathbf{v} = 2a + 1b - 4c \]

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

### Step 4: Rearrange the equation
We can rearrange the equation to express one variable in terms of the others. For example:
\[ b = 4c - 2a \]

### Step 5: Analyze the options
Now, we need to check the given options for values of \( a \), \( b \), and \( c \) to see which one satisfies the equation \( 2a + b - 4c = 0 \).

### Checking the options:
1. **Option 1:** \( a = 2, b = 3, c = -4 \)
\[ 2(2) + 3 - 4(-4) = 4 + 3 + 16 = 23 \quad (\text{not } 0) \]

2. **Option 2:** \( a = 4, b = 4, c = 5 \)
\[ 2(4) + 4 - 4(5) = 8 + 4 - 20 = -8 \quad (\text{not } 0) \]

3. **Option 3:** \( a = 4, b = 4, c = -5 \)
\[ 2(4) + 4 - 4(-5) = 8 + 4 + 20 = 32 \quad (\text{not } 0) \]

4. **Option 4:** \( a = -4, b = 4, c = -5 \)
\[ 2(-4) + 4 - 4(-5) = -8 + 4 + 20 = 16 \quad (\text{not } 0) \]

### Conclusion
None of the provided options satisfy the condition for the vectors to be perpendicular. Therefore, the answer is that none of the options are correct.