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Question
If `bar"a" = hat"i" + hat"j" + hat"k" "and" bar"c" = hat"j" - hat"k"`, find `bar"a"` vector `bar"b"` satisfying `bar"a" xx bar"b" = bar"c" "and" bar"a".bar"b" = 3`
Solution
Given: `bar"a" = hat"i" + hat"j" + hat"k" , bar"c" = hat"j" - hat"k"`
Let `bar"b" = "x"hat"i" + "y"hat"j" + "z"hat"k"`
Then `bar"a".bar"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, `bar"c" = bar"a" xx bar"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`
∴ `bar"b" = 5/3hat"i" + 2/3hat"j" + 2/3hat"k"`
i.e. `bar"b" = 1/3(5hat"i" + 2hat"j" + 2hat"k")`
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