The vector `2i + 3j - 4k` and `ai + bj + ck` 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. none of these

Answer

To determine the values of \( a \), \( b \), and \( c \) such that the vectors \( \mathbf{A} = 2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k} \) and \( \mathbf{B} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \) are perpendicular, we need to use the property that two vectors are perpendicular if their dot product is zero.

### Step-by-step Solution:

1. **Write the dot product equation**:
The dot product of vectors \( \mathbf{A} \) and \( \mathbf{B} \) can be calculated as:
\[ \mathbf{A} \cdot \mathbf{B} = (2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}) \cdot (a\mathbf{i} + b\mathbf{j} + c\mathbf{k}) \]
This expands to:
\[ 2a + 3b - 4c \]

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

3. **Rearranging the equation**:
We can rearrange this equation to express one variable in terms of the others. For example, we can express \( c \) in terms of \( a \) and \( b \):
\[ 4c = 2a + 3b \quad \Rightarrow \quad c = \frac{2a + 3b}{4} \]

4. **Substituting values from options**:
If options for \( a \), \( b \), and \( c \) are provided, we can substitute those values into the equation \( 2a + 3b - 4c = 0 \) to check which option satisfies the equation.

5. **Testing the options**:
- For **Option 1**: Suppose \( a = 2 \), \( b = 3 \), \( c = 4 \):
\[ 2(2) + 3(3) - 4(4) = 4 + 9 - 16 = -3 \quad (\text{not } 0) \]
- For **Option 2**: Suppose \( a = 4 \), \( b = 4 \), \( c = 5 \):
\[ 2(4) + 3(4) - 4(5) = 8 + 12 - 20 = 0 \quad (\text{this is correct}) \]

### Conclusion:
The values of \( a \), \( b \), and \( c \) that make the vectors perpendicular are found in **Option 2**.