Advertisement Remove all ads

If abcda¯,b¯,c¯,d¯ are four distinct vectors such that abcda¯×b¯=c¯×d¯ and acbda¯×c¯=b¯×d¯ prove that ada¯-d¯ is parallel to bcb¯-c¯. - Mathematics and Statistics

Advertisement Remove all ads
Advertisement Remove all ads
Sum

If `bar"a", bar"b", bar"c", bar"d"` are four distinct vectors such that `bar"a" xx bar"b" = bar"c" xx bar"d"` and `bar"a" xx bar"c" = bar"b" xx bar"d"` prove that `bar"a" - bar"d"` is parallel to `bar"b" - bar"c"`.

Advertisement Remove all ads

Solution

`bar"a", bar"b", bar"c", bar"d"` are four distinct vectors.

∴ `bar"a" ≠ bar"b" ≠ bar"c" ≠ bar"d"`

∴ `bar"a" - bar"d" ≠ bar"0"  "and"  bar"b" - bar"c" ≠ bar"0"`    ....(1)

Now, `bar"a" xx bar"b" = bar"c" xx bar"d"`   ...(2)

and `bar"a" xx bar"c" = bar"b" xx bar"d"`    ...(3)

Subtracting (3) from (2), we get

`bar"a" xx bar"b" - bar"a" xx bar"c" = bar"c" xx bar"d" - bar"b" xx bar"d"`

∴`bar"a" xx (bar"b" - bar"c") = (bar"c" - bar"b") xx bar"d" = - (bar"b" - bar"c") xx bar"d" = bar"d" xx (bar"b" - bar"c")`

∴ `bar"a" xx (bar"b" - bar"c") - bar"d" xx (bar"b" - bar"c") = bar"0"`

∴ `(bar"a" - bar"d") xx (bar"b" - bar"c") = bar"0"`

∴ `bar"a" - bar"d"` and `bar"b" - bar"c"` are parallel to each other.    ...[By (1)]

Concept: Vector Product of Vectors (Cross)
  Is there an error in this question or solution?
Advertisement Remove all ads
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×