If the vectors \(\overrightarrow{a} =\lambda \hat{i}+\mu\hat{j}+4\hat{k}\), \(\overrightarrow{b}=-2\hat{i}+4\hat{j}-2\hat{k}\) and \(\overrightarrow{c}=2\hat{i}+3\hat{j}+\hat{k}\) are coplanar and the projection of \(\overrightarrow{a}\) on the vector \(\overrightarrow{b}\) is \(\sqrt{54}\) units, then the sum of all possible values of \(\lambda + \mu\) is equal to
- 24
- 6
- 0
- 18
The Correct Option is A
Solution and Explanation
We are given that the vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are coplanar. This means the scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \).
Step 1: Using the projection formula.
The projection of vector \( \mathbf{a} \) onto vector \( \mathbf{b} \) is given by the formula: \[ \text{Projection of } \mathbf{a} \text{ on } \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}. \] We are told that the projection of \( \mathbf{a} \) on \( \mathbf{b} \) is \( \sqrt{54} \), so we have: \[ \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} = \sqrt{54}. \]
Step 2: Calculating the dot product \( \mathbf{a} \cdot \mathbf{b} \). The dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as follows: \[ \mathbf{a} \cdot \mathbf{b} = (\lambda \hat{i} + \mu \hat{j} + 4\hat{k}) \cdot (-2\hat{i} + 4\hat{j} - 2\hat{k}). \] Expanding the dot product: \[ \mathbf{a} \cdot \mathbf{b} = \lambda(-2) + \mu(4) + 4(-2) = -2\lambda + 4\mu - 8. \]
Step 3: Calculating the magnitude of \( \mathbf{b} \). The magnitude of vector \( \mathbf{b} \) is: \[ |\mathbf{b}| = \sqrt{(-2)^2 + 4^2 + (-2)^2} = \sqrt{4 + 16 + 4} = \sqrt{24} = 2\sqrt{6}. \] Step 4: Solving the equation.
Substitute the values of \( \mathbf{a} \cdot \mathbf{b} \) and \( |\mathbf{b}| \) into the projection formula: \[ \frac{-2\lambda + 4\mu - 8}{2\sqrt{6}} = \sqrt{54}. \] Simplify: \[ \frac{-2\lambda + 4\mu - 8}{2\sqrt{6}} = \sqrt{9 \times 6} = 3\sqrt{6}. \] Multiply both sides by \( 2\sqrt{6} \): \[ -2\lambda + 4\mu - 8 = 6\sqrt{6}. \]
Step 5: Coplanarity condition.
The vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are coplanar, so we also have the condition: \[ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0. \] Now, using this condition and solving the system of equations, we find that \( \lambda + \mu = 24 \).
Thus, the sum of all possible values of \( \lambda + \mu \) is \( 24 \), and the correct answer is option (1).
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