Question

A vector `veca` makes equal angles with all the three axes. If the magnitude of the vector is `5sqrt3` units, then find `veca`.

Answer 1

Give: If a vector `veca` makes equal angles with all three coordinate axes, then its direction cosines are equal.

Let each direction cosine be l.

∴ l = m = n

l² + m² + n² = 1

3l² = 1

l² = 1

l = `±1/sqrt3`

`|veca| = sqrt(l^2 + m^2 + n^2)`

= `sqrt(1/3 + 1/3 + 1/3)`

= 1

Unit vector in the direction of `veca`

`hata = (veca)/(|veca|) = ±(1/sqrt3 hati + 1/sqrt3  hatj + 1/sqrt3  hatk)`

∵ Magnitude is `5sqrt3`.

`|veca| = 5sqrt3`

Vector in terms of direction cosines:

`|veca| = |veca|(lhati + mhatj + nhatk)`

`veca = ±5sqrt3 (1/sqrt3  hati + 1/sqrt3  hatj + 1/sqrt3  hatk)`

`veca = ±(5hati + 5hatj + 5hatk)`