Question

A vector `vec"r"` is inclined at equal angles to the three axes. If the magnitude of `vec"r"` is `2sqrt(3)` units, find `vec"r"`.

Answer 1

Since, the vector `vec"r"` makes equal angles with the axes, their direction cosines should be same

∴ l = m = n

We know that l² + m² + n² = 1

⇒ l² + l² + l² = 1

⇒ 3l² = 1

⇒ l² =  `1/3`

⇒ l = `+- 1/sqrt(3)`

∴ `hat"r" = +- 1/sqrt(3)hat"i" +- 1/sqrt(3)hat"j" +- 1/sqrt(3)hat"k"`

⇒ `hat"k" = +- 1/sqrt(3) (hat"i" + hat"j" + hat"k")`

We know that `vec"r" = (hat"r") |vec"r"|`

= `+- 1/sqrt(3) (hat"i" + hat"j" + hat"k") 2sqrt(3)`

= `+- 2(hat"i" + hat"j" + hat"k")`

Hence, the required value of `vec"r"` is `+- 2(hat"i" + hat"j" + hat"k")`.