# If abca¯,b¯,c¯ are unit vectors such that abca¯+b¯+c¯=0¯, then find the value of abbccaa¯.b¯+b¯.c¯+c¯.a¯. - Mathematics and Statistics

Sum

If bar"a", bar"b", bar"c" are unit vectors such that bar"a" + bar"b" + bar"c" = bar0, then find the value of bar"a".bar"b" + bar"b".bar"c" + bar"c".bar"a".

#### Solution

bar"a", bar"b", bar"c" are unit vectors

∴ |bar"a"| = |bar"b"| = |bar"c"| = 1.

Also, bar"a".bar"a" = bar"b".bar"b" = bar"c".bar"c" = 1

Now, bar"a" + bar"b" + bar"c" = bar0    ...(1)

Taking scalar product of both sides with bar"a", we get

bar"a".(bar"a" + bar"b" + bar"c") = bar"a".bar0

∴ bar"a".bar"a" + bar"a".bar"b" + bar"a".bar"c" = 0

∴ bar"a".bar"b" + bar"a".bar"c" = - bar"a".bar"a" = - 1      ....(2)

Similarly taking scalar product of both sides of (1) with bar"b" and bar"c", we get,

bar"b".bar"a" + bar"b".bar"c" = - 1     ....(3)

bar"c".bar"a" + bar"c".bar"b" = - 1       .....(4)

Adding (2), (3), (4) and using the fact that scalar product is commutative, we get

2(bar"a".bar"b" + bar"b".bar"c" + bar"c".bar"a") = - 3

∴ bar"a".bar"b" + bar"b".bar"c" + bar"c".bar"a" = - 3/2`

Concept: Vectors and Their Types
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