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Question
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)]
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