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Find aa¯ if aiaj0a¯×i^+2a¯-5j^=0¯ - Mathematics and Statistics

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Sum

Find `bar"a"` if `bar"a" xx hat"i" + 2bar"a" - 5hat"j" = bar"0"`

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Solution

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"`

Concept: Vector Product of Vectors (Cross)
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