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Question
Dot product of a vector with vectors `3hat"i" - 5hat"k", 2hat"i" + 7hat"j" and hat"i" + hat"j" + hat"k"` are respectively -1, 6 and 5. Find the vector.
Solution
Let `bar"a" = 3hat"i" - 5hat"k", bar"b" = 2hat"i" + 7hat"j", bar"c" = hat"i" + hat"j" + hat"k"`
Let `bar"r" = "x"hat"i" + "y"hat"j" + "z"hat"k"` be the required vector.
Then, `bar"r".bar"a" = -1, bar"r".bar"a" = 6, bar"r".bar"c" = 5`
∴ `("x"hat"i" + "y"hat"j" + "z"hat"k").(3hat"i" - 5hat"k") = - 1`
`("x"hat"i" + "y"hat"j" + "z"hat"k").(2hat"i" + 7hat"j") = 6` and
`("x"hat"i" + "y"hat"j" + "z"hat"k").(hat"i" + hat"j" + hat"k") = 5`
∴ 3x - 5y = - 1 ....(1)
∴ 2x + 7y = 6 ....(2)
∴ x + y + z = 5 ....(3)
From (3), z = 5 - x - y
Substituting this value of z in (1), we get
∴ 3x - 5(5 - x - y) = - 1
∴ 8x + 5y = 24 ....(4)
Multiplying (2) by 4 and subtracting from (4), we get
8x + 5y - 4(2x + 7y) = 24 - 6 × 4
∴ - 23y = 0
∴ y = 0
Substituting y = 0 in (2), we get
∴ 2x = 6
∴ x = 3
Substituting x = 3 in (1), we get
∴ 3(3) - 5z = - 1
∴ - 5z = - 10
∴ z = 2
∴ `bar"r" = 3hat"i" + 0.hat"j" + 2hat"k"`
`= 3hat"i" + 2hat"k"`
Hence, the required vector is `3hat"i" + 2hat"k"`.
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