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प्रश्न
Find a unit vector perpendicular to the plane containing the point (a, 0, 0), (0, b, 0) and (0, 0, c). What is the area of the triangle with these vertices?
उत्तर
The position vectors `bar"p", bar"q", bar"r"` of the points A(a, 0, 0), B(0, b, 0), C(0, 0, c) are
`bar"p" = "a"hat"i", bar"q" = "b"hat"j", bar"r" = "c"hat"k"`
`bar"AB" = bar"q" - bar"p" = "b"hat"j" - "a"hat"i" = - "a"hat"j" + "b"hat"j"`
`bar"BC" = bar"r" - bar"q" = "c"hat"k" - "b"hat"j" = - "b"hat"j" + "c"hat"k"`
`bar"AB" xx bar"BC" = |(hat"i",hat"j",hat"k"),(-"a","b",0),(0,-"b","c")|`
`= ("bc" - 0)hat"i" - (- "ac" - 0)hat"j" + ("ab" - 0)hat"k"`
`= "bc"hat"i" + "ac"hat"j" + "ab"hat"k"`
`|bar"AB" xx bar"BC"| = sqrt(("bc")^2 + ("ac")^2 + ("ab")^2)`
`= sqrt("b"^2"c"^2 + "a"^2"c"^2 + "a"^2"b"^2)`
`bar"AB" xx bar"BC"` is perpendicular to the plane containing A, B, C.
∴ the required unit vector
`= (bar"AB" xx bar"BC")/(|bar"AB" xx bar"BC"|) = ("bc"hat"i" + "ca"hat"j" + "ab"hat"k")/sqrt("b"^2"c"^2 + "c"^2"a"^2 + "a"^2"b"^2)`
Area of Δ ABC = `1/2 |bar"AB" xx bar"BC"|`
`= 1/2 sqrt("b"^2"c"^2 + "a"^2"c"^2 + "a"^2"b"^2)` sq.units.
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