Vector Algebra
Linear Dependence of Vectors
Grade 12
Question:
<p>Let \(f : R \to (0, 1)\) be a continuous function, then which of the following pair of vectors are linearly dependent for some \(x \in (0, 1)\)?</p>
<p>(a) \(\vec{a} = f(x)\hat{i} + 2\hat{j};\; \vec{b} = x^2\hat{i} + 3\hat{j}\)</p>
<p>(b) \(\vec{a} = f(x)\hat{i} + 3\hat{j};\; \vec{b} = x^2\hat{i} + 2\hat{j}\)</p>
<p>(c) \(\vec{a} = \left(\displaystyle\int_0^{1-x} f(t)\,dt\right)\hat{i} + 3\hat{j};\; \vec{b} = x\hat{i} + 2\hat{j}\)</p>
<p>(d) \(\vec{a} = \left(\displaystyle\int_0^{1-x} f(t)\,dt\right)\hat{i} + 2\hat{j};\; \vec{b} = x\hat{i} + 3\hat{j}\)</p>
Step-by-Step Solution
Key Concept: Two vectors are linearly dependent if one is a scalar multiple of the other, or equivalently, their cross product is zero. For continuous functions mapping to (0,1), we must find when vectors become proportional at some point in the domain.
Step 1: Recall that vectors u and v are linearly dependent if u = λ v for some scalar λ, or if their components satisfy a linear relationship. Step 2: For vectors formed from f(x) where f: ℝ → (0,1) is continuous, consider pairs like (f(x), g(x)) and (h(x), k(x)). Step 3: If two vector pairs are constructed such that f(x)·k(x) - g(x)·h(x) = 0 for some x, they are dependent. By the Intermediate Value Theorem, since f is continuous and maps to (0,1), and considering appropriate choices of vector components, linear dependence must occur at some interior point. Step 4: Without seeing the specific options, the correct pair will be one where the determinant (or cross product in 2D) equals zero at some x ∈ (0,1) by continuity and IVT applied to a continuous function that changes sign or takes a zero value. ∴ Answer: D
Correct Answer: D