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Ex. prove that the vectors 2i+j-3k; i -4k and 4i + 3j-k are linearly dependent sol. the vectors a, b, c, ... are called linearly dependent if we can find a set
Ex. Prove that the vectors 2i+j-3k; i -4k and 4i + 3j-k are linearly dependent Sol. The vectors A, B, C, ... are called linearly dependent if we can find a set of scalars a, b, c, .. not all zera that aA + bB +cC+=0; otherwise they are linearly independent. Apply this rule for given vectors, a(2i+j-3k) + bi - 4k)+c(4i +3j-k) =0 (2a + b +4c)i + (a + 3c) j + (-3a 4b c)k =0 or Since i, j, k are non-coplanar, 2a +b + 4c = 0 a + 3c = 0 and 3a + 4b + c = 0 Solving above equations, we get a = - 3, b = 2, c =1 1 hus equation (1) is satisfied for non-zero a, b and c and hence given vectors are linearly dependent.
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