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Consider the following statements: Statement 1: If vectors a⃗ = 2î + k̂, b⃗ = 3ĵ + 4k̂, and c⃗ = 8î − 3ĵ are coplanar, then c⃗ = 4a⃗ − b⃗. Statement 2: A collection of vectors a⃗1, a⃗2, a⃗3, ..., a⃗n is called linearly independent if the equation ℓ1a⃗1 + ℓ2a⃗2 + ... + ℓna⃗n = 0 can hold only when ℓ1 = ℓ2 = ... = ℓn = 0, where the ℓ's are scalars. Choose the correct option:

1. Statement-1 is true. Statement-2 is true. Statement-2 is a correct explanation for Statement-1
2. Statement-1 is True, Statement-2 is True : Statement-2 is NOT a correct explanation for Statement-1
3. Statement-1 is False, Statement-2 is True
4. Statement-1 is True, Statement-2 is False

Correct answer: Statement-1 is True, Statement-2 is True : Statement-2 is NOT a correct explanation for Statement-1

Solution

Statement-1 is true because the given vectors can be expressed in terms of each other, confirming their coplanarity. Statement-2 accurately defines linear independence, but it does not directly explain the coplanarity condition of the vectors in Statement-1.

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