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प्रश्न
Using properties of scalar triple product, prove that `[(bar"a" + bar"b", bar"b" + bar"c", bar"c" + bar"a")] = 2[(bar"a", bar"b", bar"c")]`.
उत्तर
L.H.S = `[(bar"a" + bar"b", bar"b" + bar"c", bar"c" + bar"a")]`
`= (bar"a" + bar"b").[(bar"b" + bar"c") xx (bar"c" + bar"a")]`
`= (bar"a" + bar"b").[bar"b" xx bar"c" + bar"b" xx bar"a" + bar"c" xx bar"c" + bar"c" xx bar"a"]`
`= (bar"a" + bar"b").[bar"b" xx bar"c" + bar"b" xx bar"a" + bar"c" xx bar"a"] ....[∵ bar"c" xx bar"c" = bar"0"]`
`= bar"a".[(bar"b" xx bar"c") + (bar"b" xx bar"a") + (bar"c" xx bar"a")] + bar"b".[(bar"b" xx bar"c") + (bar"b" xx bar"a") + (bar"c" xx bar"a")]`
`= bar"a".(bar"b" xx bar"c") + bar"a".(bar"b" xx bar"a") + bar"a".(bar"c" xx bar"a") + bar"b".(bar"b" xx bar"c") + bar"b"(bar"b" xx bar"a") + bar"b"(bar"c" xx bar"a")`
`= [bar"a" bar"b" bar"c"] + [bar"a" bar"b" bar"a"] + [bar"a" bar"c" bar"a"] + [bar"b" bar"b" bar"c"] + [bar"b" bar"b" bar"a"] + [bar"b" bar"c" bar"a"]`
`= [bar"a" bar"b" bar"c"] + 0 + 0 + 0 + 0 + [bar"a" bar"b" bar"c"] `
`= 2[bar"a" bar"b" bar"c"]`
= R.H.S
संबंधित प्रश्न
If A, B, C, D are (1, 1, 1), (2, 1, 3), (3, 2, 2), (3, 3, 4) respectively, then find the volume of parallelopiped with AB, AC and AD as the concurrent edges.
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(A) 100
(B) 101
(C) 110
(D) 109
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