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Dot product of a vector with vectors ikijandijk3i^-5k^, 2i^+7j^andi^+j^+k^ are respectively -1, 6 and 5. Find the vector. - Mathematics and Statistics

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Sum

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.

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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"`.

Concept: Vectors and Their Types
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