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Question
If `vec"a" = hat"i" + hat"j" + hat"k"` and `vec"c" = hat"j" - hat"k"`. find a vector `vec"b"` satisfying `vec"a" xx vec"b" = vec"c"` and `vec"a"·vec"b"` = 3.
Solution
Given, `vec"a" = hat"i" + hat"j" + hat"k"`, `vec"c" = hat"j" - hat"k"`
Let `vec"b" = "x"hat"i"+"y"hat"j"+"z"hat"k"`
Then `vec"a"·vec"b"` = 3 gives
`(hat"i" + hat"j" + hat"k"). ("x"hat"i"+"y"hat"j"+"z"hat"k")`= 3
:. (1)(x) + (1)(y) + (1)(z) = 3
Also, x + y + z = 3 ...(1)
Also, `vec"c" = hat"a"xxhat"b"`
`hat"j" - hat"k" = |(hat"i", hat"j", hat"k"),(1, 1, 1),("x", "y", "z")|`
= (z - y)`hat"i"` - (z - x)`hat"j"` + (y - x)`hat"k"`
= (z - y)`hat"i"` + (x - z)`hat"j"` + (y - x)`hat"k"`
By equality of vectors,
z - y = 0 ...(2)
x - z = 1 ...(3)
y - x = - 1 ...(4)
From (2), y = z
From (3), x = 1 + z
Substituting these values of x and y in (1), we get
1 + z + z + z = 3
`"z" = 2/3`
y = z = `2/3`
x = 1 + z = 1 + `2/3` = `5/3`
`vec"b" 5/3hat"i" + 2/3hat"j" +2/3hat"k"`
i.e, `vec"b" = 1/3(5hat"i" + 2hat"j" + 2hat"k")`
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