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Statement I: If a=2hati+hatk,b=3hatj+4ha...

Statement I: If `a=2hati+hatk,b=3hatj+4hatk and c=lamda a+mub` are coplanar, then `c=4a-b`.
Statement II: A set vector `a_(1),a_(2),a_(3), . . ,a_(n)` is said to be linearly independent, if every relation of the form
`l_(1)a_(1)+l_(2)a_(2)+l_(3)a_(3)+ . . .+l_(n)a_(n)=0` implies that `l_(1)=l_(2)=l_(3)= . . .=l_(n)=0` (scalar).

A

Statement-I and statement II ar correct and Statement II is the correct explanation of statement I

B

Both statement I and statement II are correct but statement II is not the correct explanation of statement I

C

Statement I is correct but statement II is incorrect

D

Statement II is correct but statement I is incorrect

Text Solution

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The correct Answer is:
To solve the problem step by step, we will analyze the statements and verify their correctness. ### Step 1: Understand the Vectors We are given three vectors: - \( \mathbf{a} = 2\hat{i} + \hat{k} \) - \( \mathbf{b} = 3\hat{j} + 4\hat{k} \) - \( \mathbf{c} = \lambda \mathbf{a} + \mu \mathbf{b} \) ### Step 2: Determine the Condition for Coplanarity Vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are coplanar if their scalar triple product is zero. The scalar triple product can be computed using the determinant of a matrix formed by the vectors. ### Step 3: Set Up the Determinant The scalar triple product can be represented as: \[ \begin{vmatrix} 2 & 0 & 1 \\ 0 & 3 & 4 \\ 2\lambda & 3\mu & \lambda + 4\mu \end{vmatrix} = 0 \] ### Step 4: Calculate the Determinant Expanding the determinant: \[ = 2 \begin{vmatrix} 3 & 4 \\ 3\mu & \lambda + 4\mu \end{vmatrix} - 0 + 1 \begin{vmatrix} 0 & 3 \\ 2\lambda & 3\mu \end{vmatrix} \] Calculating the first determinant: \[ = 2(3(\lambda + 4\mu) - 4(3\mu)) = 2(3\lambda + 12\mu - 12\mu) = 6\lambda \] Calculating the second determinant: \[ = 0 - 6\lambda = -6\lambda \] Combining these results: \[ 6\lambda - 6\lambda = 0 \] ### Step 5: Conclusion on Coplanarity The determinant simplifies to \( 0 = 0 \), which is true for all values of \( \lambda \) and \( \mu \). Therefore, vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are coplanar for any values of \( \lambda \) and \( \mu \). ### Step 6: Check the Statement To verify statement I, we need to check if \( \mathbf{c} = 4\mathbf{a} - \mathbf{b} \) is valid for some \( \lambda \) and \( \mu \): Assuming \( \lambda = 4 \) and \( \mu = -1 \): \[ \mathbf{c} = 4(2\hat{i} + \hat{k}) - (3\hat{j} + 4\hat{k}) = 8\hat{i} + 4\hat{k} - 3\hat{j} - 4\hat{k} = 8\hat{i} - 3\hat{j} \] This confirms that \( \mathbf{c} = 4\mathbf{a} - \mathbf{b} \) is indeed valid. ### Step 7: Analyze Statement II Statement II defines linear independence. A set of vectors is linearly independent if the only solution to the equation \( l_1\mathbf{a}_1 + l_2\mathbf{a}_2 + \ldots + l_n\mathbf{a}_n = 0 \) is \( l_1 = l_2 = \ldots = l_n = 0 \). This is a true statement about linear independence. ### Final Conclusion Both statements are true, but Statement II does not provide a correct explanation for Statement I, as they address different concepts (coplanarity vs. linear independence). ### Summary - **Statement I**: True - **Statement II**: True, but not a correct explanation of Statement I.
To solve the problem step by step, we will analyze the statements and verify their correctness. ### Step 1: Understand the Vectors We are given three vectors: - \( \mathbf{a} = 2\hat{i} + \hat{k} \) - \( \mathbf{b} = 3\hat{j} + 4\hat{k} \) - \( \mathbf{c} = \lambda \mathbf{a} + \mu \mathbf{b} \) ...
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