# 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

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".

#### 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)
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