The equation of a line is 5x = 15y + 7 = 3z. Write the direction cosines of the line.

5x = 15y + 7 = 3z → x = (y - (7/15)) / (1/5) = z / (5/3).
Comparing this with the standard equation of straight line: (x - a1)/b1 = (y - a2)/b2 = (z - a3)/b3, we get, b1 = 1/5, b2 = 1/15, b3 = 5/3.
The direction cosines are the components of the unit vector
^b = b1^i + b2^j + b3^k
√(b1² + b2² + b3²) = √((1/5)² + (1/15)² + (5/3)²) = √(1/25 + 1/225 + 25/9) = √(9/225 + 1/225 + 625/225) = √(635/225) = √635/15
= (1/5)^i + (1/15)^j + (5/3)^k / (√635/15) = (3/√635)^i + (1/√635)^j + (25/√635)^k
So, the direction cosines are 3/√635, 1/√635, 25/√635.