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प्रश्न
Find `bar"a"` if `bar"a" xx hat"i" + 2bar"a" - 5hat"j" = bar"0"`
उत्तर
Let `bar"a" = "x"hat"i" + "y"hat"j" + "z"hat"k"`
Then `bar"a" xx hat"i" = ("x"hat"i" + "y"hat"j" + "z"hat"k") xx hat"i"`
`= "x"(hat"i" xx hat"i") + "y"(hat"j" xx hat"i") + "z"(hat"k" xx hat"i")`
`= "z"hat"j" - "y"hat"k"` ....`[∵ hat"i" xx hat"i" = hat"0", hat"j" xx hat"i" = - hat"k", hat"k"xx hat"i" = hat"j"]`
It is given that
`bar"a" xx hat"i" + 2bar"a" - 5hat"j" = bar"0"`
∴ `"z"hat"j" - "y"hat"k" + 2("x"hat"i" + "y"hat"j" + "z"hat"k") - 5hat"j" = 0`
∴ `"z"hat"j" - "y"hat"k" + 2"x"hat"i" + 2"y"hat"j" + 2"z"hat"k" - 5hat"j" = bar"0"`
∴ `2"x"hat"i" + (2"y" + "z" - 5)hat"j" + (2"z" - "y")hat"k" = bar"0"`
By equality of vectors
2x = 0 i.e. x = 0
2y + z - 5 = 0 ....(1)
2z - y = 0 ....(2)
From (2), y = 2z
Substituting y = 2z in (1), we get
4z + z = 5
∴ z = 1
∴ y = 2z = 2(1) = 2
∴ x = 0, y = 2, z = 1
∴ `bar"a" = 2hat"j" + hat"k"`
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