Vector Algebra
Coplanar vectors and linear combinations
Grade 12
Question:
<p><strong>Ex. 49 Statement I:</strong> If \(\mathbf{a} = 2\mathbf{i} + \mathbf{k}\), \(\mathbf{b} = 3\mathbf{j} + 4\mathbf{k}\) and \(\mathbf{c} = \lambda \mathbf{a} + \mu\mathbf{b}\) are coplanar, then \(\mathbf{c} = 4\mathbf{a} - \mathbf{b}\).</p><p><strong>Statement II:</strong> A set of vectors \(\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3, \ldots, \mathbf{a}_n\) is said to be linearly independent, if every relation of the form \(l_1\mathbf{a}_1 + l_2\mathbf{a}_2 + l_3\mathbf{a}_3 + \cdots + l_n\mathbf{a}_n = 0\) implies that \(l_1 = l_2 = l_3 = \cdots = l_n = 0\) (scalar).</p>
<p>(a) Both Statement I and Statement II are correct and Statement II is the correct explanation of Statement I</p>
<p>(b) Both Statement I and Statement II are correct but Statement II is not the correct explanation of Statement I</p>
<p>(c) Statement I is correct but Statement II is incorrect</p>
<p>(d) Statement II is correct but Statement I is incorrect</p>
Step-by-Step Solution
Key Concept: Three vectors are coplanar if one can be expressed as a linear combination of the other two; linear independence is a related but distinct concept.
\(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) are coplanar, so \(\mathbf{c} = \lambda \mathbf{a} + \mu\mathbf{b}\). Solving with the given vectors: \(\lambda = 4\) and \(\mu = -1\). Hence \(\mathbf{c} = 4\mathbf{a} - \mathbf{b}\). Statement II correctly defines linear independence. Both statements are correct but Statement II does not explain Statement I. ∴ Answer is (b).
Correct Answer: b