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Question: The value of [**a** – **b** **b** – **c** **c** – **a**], where \(|a| = 1\), \(|b| = 5\) and \(|c| =...

The value of [a – b b – c c – a], where ∣a∣=1|a| = 1, ∣b∣=5|b| = 5 and ∣c∣=3|c| = 3 is

A

0

B

1

C

2

D

4

Answer

0

Explanation

Solution

[a−bb−cc−a]={(a−b)×(b−c)}.(c−a)\lbrack a - bb - cc - a\rbrack = \{(a - b) \times (b - c)\}.(c - a)

=(a×b−a×c−b×b+b×c).(c−a)= (a \times b - a \times c - b \times b + b \times c).(c - a)

=(a×ab+ca×a+b×c).(c−a)= (a \times ab + ca \times a + b \times c).(\mathbf{c} - \mathbf{a})

=(a×b).c−(a×b).a+(c×a).c−(c×a).a= (a \times b).c - (a \times b).a + (c \times a).c - (c \times a).a

= (a×b).c−(a×b).a+(c×a).c−(c×a).a+(b×c).c−(b×c).a(\mathbf{a} \times \mathbf{b}).\mathbf{c} - (\mathbf{a} \times \mathbf{b}).\mathbf{a} + (\mathbf{c} \times \mathbf{a}).\mathbf{c} - (\mathbf{c} \times \mathbf{a}).\mathbf{a} + (\mathbf{b} \times \mathbf{c}).\mathbf{c} - (\mathbf{b} \times \mathbf{c}).\mathbf{a}

=[abc]−[aba]+[cac]−[caa]+[bcc]−[bca]= \lbrack abc\rbrack - \lbrack aba\rbrack + \lbrack cac\rbrack - \lbrack caa\rbrack + \lbrack bcc\rbrack - \lbrack bca\rbrack = 0.