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प्रश्न
Find two unit vectors each of which makes equal angles with bar"u", bar"v" and bar"w" where bar"u" = 2hat"i" + hat"j" - 2hat"k", bar"v" = hat"i" + 2hat"j" - 2hat"k", bar"w" = 2hat"i" - 2hat"j" + hat"k".
उत्तर
Let `bar"r" = "x"hat"i" + "y"hat"j" + "z"hat"k"` be the unit vector which makes angle θ with each of the vectors
Then `|bar"r"| = 1`
Also, `bar"u" = 2hat"i" + hat"j" - 2hat"k", bar"v"= hat"i" + 2hat"j" - 2hat"k", bar"w" = 2hat"i" - 2hat"j" + hat"k"`
`|bar"u"| = sqrt(2^2 + 1^2 + (- 2)^2) = sqrt(4 + 1 + 4) = sqrt9 = 3`
`|bar"v"| = sqrt(1^2 + 2^2 + (- 2)^2) = sqrt(1 + 4 + 4) = sqrt9 = 3`
`|bar"w"| = sqrt(2^2 + (- 2)^2 + 1^2) = sqrt(4 + 4 + 1) = sqrt9 = 3`
Angle between `bar"r" and bar"u"` is θ
∴ cos θ = `(bar"r".bar"u")/(|bar"r"||bar"u"|)`
`= (("x"hat"i" + "y"hat"j" + "z"hat"k").(2hat"i" + hat"j" - 2hat"k"))/(1xx3)`
`= (2"x" + "y" - 2"z")/3` ....(1)
Also, the angle between `bar"r" and bar"v"` and between `bar"r" and bar"w"` is θ.
∴ cos θ = `(bar"r".bar"u")/(|bar"r"||bar"u"|)`
`= (("x"hat"i" + "y"hat"j" + "z"hat"k").(hat"i" + 2hat"j" - 2hat"k"))/(1xx3)`
`= ("x" + 2"y" - 2"z")/3` ....(2)
and cos θ `= (bar"r".bar"u")/(|bar"r"||bar"u"|)`
`= (("x"hat"i" + "y"hat"j" + "z"hat"k").(2hat"i" - 2hat"j" + hat"k"))/(1xx3)`
`= (2"x" - 2"y" + "z")/3` ....(3)
From (1) and (2), we get
`(2"x" + "y" - 2"z")/3 = ("x" + 2"y" - 2"z")/3`
∴ 2x + y - 2z = x + 2y - 2z
∴ x = y
From (2) and (3), we get
`("x" + 2"y" - 2"z")/3 = (2"x" - 2"y" + "z")/3`
∴ x + 2y - 2z = 2x - 2y + z
∴ 3y = 3z .....[∵ x = y]
∴ y = z
∴ x = y = z
∴ `bar"r" = "x"hat"i" + "y"hat"j" + "z"hat"k" = "x"hat"i" + "x"hat"j" + "x"hat"k"`
∴ `|bar"r"| = sqrt("x"^2 + "x"^2 + "x"^2) = 1`
∴ `"x"^2 + "x"^2 + "x"^2 = 1`
∴ `3"x"^2 = 1`
∴ `"x"^2 = 1/3`
∴ x = `- 1/sqrt3 `
∴ `bar"r" = +- 1/sqrt3 hat"i" +- 1/sqrt3 hat"j" +- 1/sqrt3hat"k" `
`= +- 1/sqrt3 (hat"i" + hat"j" + hat"k")`
Hence, the required unit vectors are `+- 1/sqrt3 (hat"i" + hat"j" + hat"k")`
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