If the vectors $\overrightarrow a = \widehat i - \widehat j + 2\widehat k,\,\,\,\,\,\overrightarrow b = 2\widehat i + 4\widehat j + \widehat k\,\,\,$ and $\,\overrightarrow c = \lambda \widehat i + \widehat j + \mu \widehat k$ are mutually orthogonal, then $\,\left( {\lambda ,\mu } \right)$ is equal to :
$(2, -3)$
$(-2, 3)$
$(3, -2)$
$(-3, 2)$
Solution
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Step-by-Step Solution
Step 1: Understand the Orthogonality Condition
For three vectors $ \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} $ to be mutually orthogonal, each pair of vectors must have a zero dot product. That is:
$ \overrightarrow{a} \cdot \overrightarrow{b} = 0, \quad
\overrightarrow{b} \cdot \overrightarrow{c} = 0, \quad
\overrightarrow{c} \cdot \overrightarrow{a} = 0 $
Step 2: Identify the Components of the Vectors
Rewrite the given vectors in component form:
$ \overrightarrow{a} = (1,\, -1,\, 2), \quad
\overrightarrow{b} = (2,\, 4,\, 1), \quad
\overrightarrow{c} = (\lambda,\, 1,\, \mu). $
We already know $ \overrightarrow{a} \cdot \overrightarrow{b} = 0 $ is satisfied by the given vectors.
Step 3: Set Up the Orthogonality Conditions Involving $ \overrightarrow{c} $
Since $ \overrightarrow{b} $ and $ \overrightarrow{c} $ are orthogonal:
$ \overrightarrow{b} \cdot \overrightarrow{c} = (2)(\lambda) + (4)(1) + (1)(\mu) = 2\lambda + 4 + \mu = 0. $
This simplifies to:
$ 2\lambda + \mu = -4. \quad \text{(Equation 1)}
Since $ \overrightarrow{c} $ and $ \overrightarrow{a} $ are orthogonal:
$ \overrightarrow{c} \cdot \overrightarrow{a} = \lambda (1) + (1)(-1) + \mu (2) = \lambda - 1 + 2\mu = 0.
This simplifies to:
$ \lambda + 2\mu = 1. \quad \text{(Equation 2)}
Step 4: Solve the System of Equations
We have two linear equations:
1) $ 2\lambda + \mu = -4 $
2) $ \lambda + 2\mu = 1 $
From Equation (1): $ \mu = -4 - 2\lambda. $
Substitute this into Equation (2):
$ \lambda + 2(-4 - 2\lambda) = 1. $
$ \lambda - 8 - 4\lambda = 1. $
$ -3\lambda = 9. $
$ \lambda = -3. $
Plug $ \lambda = -3 $ back into $ \mu = -4 - 2\lambda $:
$ \mu = -4 - 2(-3) = -4 + 6 = 2. $
Step 5: Conclude the Values of $ \lambda $ and $ \mu $
The ordered pair satisfying both orthogonality conditions is:
$ (\lambda,\, \mu) = (-3,\, 2). $
Hence, the correct answer is $(-3, 2)$.
