# 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" - Mathematics and Statistics

Sum

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".

#### Solution

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")

Concept: Vectors and Their Types
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