Consider the following statements:

Statement 1: If vectors a = 2î + k̂, b = 3ĵ + 4k̂, and c = 8î − 3ĵ are coplanar, then c = 4a − b.

Statement 2: A collection of vectors a1, a2, a3,..., an is called linearly independent if the equation ℓ1a1 + ℓ2a2 +... + ℓnan = 0 can hold only when ℓ

Question

Consider the following statements:

Statement 1: If vectors a = 2î + k̂, b = 3ĵ + 4k̂, and c = 8î − 3ĵ are coplanar, then c = 4a − b.

Statement 2: A collection of vectors a1, a2, a3,..., an is called linearly independent if the equation ℓ1a1 + ℓ2a2 +... + ℓnan = 0 can hold only when ℓ1 = ℓ2 =... = ℓn = 0, where the ℓ's are scalars.

Choose the correct option:

Accepted Answer

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

Solution

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