If the direction cosines of a straight line are `l, m and n`,then prove that `l^2 + m^2 + n^2 = 1`.

Explanation

Let `alpha, beta , gamma` be the angle made by the line with `X-, Y-, Z-` axes respectively.
`therefore l = cos alpha, m = cos beta` and `n = cos gamma`
Let `bara = a_(1)hati + a_(2)hatj + a_(3)hatk` be any non-zero vector along the line.
Since `hati` is the unit vector along X-axis,
`bara*hati = |bara|*|hati|cos alpha= a cos alpha`
Also, `bara * hati = (a_(1)hati + a_(2)hatj + a_(3)hatk) * hati`
` = a_(1) xx 1 + a_(2) xx 0 + a_(3) xx 0 = a_(1)`
`therefore a cos alpha = a_(1) " "...(1)`
Since `hatj` is the unit vector along Y-axis,
`bara*hatj = |bara|*|hatj| cos beta = a cos beta`
Also, `bara * hatj = (a_(1)hati + a_(2)hatj + a_(3)hatk)*hatj`
` =a_(1) xx 0 + a_(2) xx 1 + a_(3) xx 0 = a_(2)`
`therefore a cos beta = a_(2)" "...(2)`
Similarly, `a cos gamma = a_(3)" "...(3)`
`therefore` from (equations (1), (2) and (3),
`a^(2)cos^(2)alpha+a^(2)cos^(2)beta+a^(2)cos^(2)gamma=1a_(1)^(2)+a_(2)^(2)+a_(3)^(2)`
`therefore a^(2)(cos^(2)alpha+cos^(2)beta+cos^(2)gamma)=a^(2)" "...[becausea =|bara|=sqrt(a_(1)^(2)+a_(2)^(2)+a_(3)^(2))]`
`therefore cos^(2)alpha+cos^(2)beta+cos^(2)gamma=1" "...[becauseaneo0]`
i.e., `l^(2) + m^(2) + n^(2) = 1`.
Now, `sin^(2)alpha + sin^(2) beta + sin^(2)gamma`
` = 1 - cos^(2) alpha + 1 - cos^(2) beta + 1 - cos^(2) gamma`
` = 3 - (cos^(2) alpha + cos^(2) beta + cos^(2)gamma)`
` = 3- 1" "...[because cos^(2) alpha + cos^(2) beta + cos ^(2) gamma = 1]`
` = 2.`