Project ForgeEngineering Mathematics
Linear Algebra
GATE 2014
Medium
Which one of the following describes the relationship among the three vectors, î+ĵ+k̂, 2î+3ĵ+k̂ and 5î+6ĵ+4k̂?
- AThe vectors are mutually perpendicular
- The vectors are linearly dependent
- CThe vectors are linearly independent
- DThe vectors are unit vectors
Explanation
Check whether the third vector is a linear combination of the first two: 3(î+ĵ+k̂) + 1(2î+3ĵ+k̂) = (3+2)î+(3+3)ĵ+(3+1)k̂ = 5î+6ĵ+4k̂ — exactly the third vector. A non-trivial linear combination of the first two equals the third, so all three are linearly dependent.
Source: GATE ME 2014 (Session 1 / Set-1) (2014)
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