Question

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".

Answer 1

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")`