Direction Ratios and Direction Cosines:

Direction ratios and direction cosines are used to describe the direction of a line in three-dimensional space.

Direction ratios are the numerical values that represent the ratios of the displacements along the x, y, and z axes. They can be denoted as (a, b, c), where a is the ratio of displacement along the x-axis, b is the ratio of displacement along the y-axis, and c is the ratio of displacement along the z-axis.

Direction cosines, on the other hand, are the cosines of the angles that a line makes with the positive x, y, and z axes. They can be denoted as (l, m, n), where l is the cosine of the angle between the line and the x-axis, m is the cosine of the angle between the line and the y-axis, and n is the cosine of the angle between the line and the z-axis.

Relationship between Direction Ratios and Direction Cosines:

There is a relationship between direction ratios and direction cosines. If (a, b, c) are the direction ratios of a line, and (l, m, n) are the direction cosines of the line, then the following relationship holds true:

l = a / √(a^2 + b^2 + c^2)
m = b / √(a^2 + b^2 + c^2)
n = c / √(a^2 + b^2 + c^2)

Finding Direction Cosines:

In this question, the direction ratios of the line are given as -4, 18, -12. To find the direction cosines, we can use the above relationship.

a = -4, b = 18, c = -12

Calculating the magnitude of the direction ratios:
√(a^2 + b^2 + c^2) = √((-4)^2 + 18^2 + (-12)^2) = √(16 + 324 + 144) = √484 = 22

Now, substituting the values of a, b, c, and the magnitude into the formulas for the direction cosines:

l = -4 / 22 = -2/11
m = 18 / 22 = 9/11
n = -12 / 22 = -6/11

Therefore, the direction cosines of the given line are -2/11, 9/11, -6/11.

Answer: d) -2/11, 9/11, -6/11

Direction ratio are -4,18,-12 so dividing by√-4^2 +18^2+-12^2 .so l=-4/22 ,m=18/22 ,n=-12/22. ,-2/11,9/11 ,-6/11.