A line makes angles of measures `45^(@)` and `60^(@)` with positive directions of Y-and Z-axes respectively. Find the d.c.s. of the line and also find the vector of magnitude 5 along the direction of line.

Explanation

Let `alpha , beta ` and `gamma` be the angles made by the line with positive directions of X, Y and Z axes.
Let l, m, n be the direction cosines of the line. Then `a=cos alpha, m = cos beta, n = cos gamma`
Here , `beta = 45^(@) ` and ` gamma=60^(@)`
`:. m = cos beta = cos 45^(@)=(1)/(sqrt(2))`
`n=cos gamma = cos 60^(@)=(1)/(2)`
We have `cos^(2) alpha + cos^(2)beta+cos^(2) gamma = 1`
`:.cos^(2)alpha+1/2+1/4=1`
`:.cos^(2)alpha=1/4`
`:.cos alpha = pm 1/2`
`rArr l=pm 1/2`
`:.` The direction cosines of the lines are
`1/2, 1/sqrt(2), 1/2` and `(-1)/(2),(1)/(sqrt(2)), (1)/(2)`.
`:.` The unit vectors along the direction of line are
`(1)/(2) hat(i) + (1)/(sqrt(2))hat(j)+(1)/(2) hat(k) ` and `((-1)/(2)) hat(i) + (1)/(sqrt(2)) hat(j)+(1)/(2) hat(k)`
Now, `|a|=5`
`:' bar(a)=5(1/2 hat(i)+(1)/(sqrt(2))hat(j)+(1)/(2) hat(k))`
And `bar(a)=5((-1)/(2)hat(i)+(1)/(sqrt(2))hat(j)+(1)/(2) hat(k))`