Advertisement Remove all ads

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

Advertisement Remove all ads
Advertisement Remove all ads
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".`

Advertisement Remove all ads

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
  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×