Exercise 11.1 — Three Dimensional Geometry

1. 13 marksNCERT Exercise

   If a line makes angles 90 degrees, 135 degrees, 45 degrees with the x, y and z-axes respectively, find its direction cosines.

   Hint. Direction cosines are just the cosines of the three direction angles.

   Since the direction cosines are cos(90), cos(135), and cos(45), evaluating each gives 0, -1/sqrt2, and 1/sqrt2 respectively.

   ✦ (0, -1/sqrt(2), 1/sqrt(2))

2. 23 marksNCERT Exercise

   Find the direction cosines of a line which makes equal angles with the coordinate axes.

   Hint. If all three angles are equal, all three direction cosines are equal; use l^2+m^2+n^2=1.

   Since equal angles mean l=m=n=t, and every set of direction cosines satisfies l^2+m^2+n^2=1, we get 3t^2=1, so t=+-1/sqrt3 for all three simultaneously.

   ✦ (+-1/sqrt(3), +-1/sqrt(3), +-1/sqrt(3))

3. 33 marksNCERT Exercise

   If a line has the direction ratios -18, 12, -4, then what are its direction cosines?

   Hint. Divide each direction ratio by the square root of the sum of their squares.

   Since sqrt(324+144+16)=sqrt484=22, dividing each direction ratio by 22 gives the direction cosines.

   ✦ (-9/11, 6/11, -2/11)

4. 43 marksNCERT Exercise

   Show that the points (2,3,4), (-1,-2,1), (5,8,7) are collinear.

   Hint. Form the direction ratios of two segments and check whether one set is proportional to the other.

   Since the direction ratios of AB are (-3,-5,-3) and those of BC are (6,10,6), and BC is exactly -2 times AB, the two segments are parallel; because they share the common point B, all three points lie on one line.

   ✦ Collinear, since the direction ratios of BC are -2 times those of AB

5. 53 marksNCERT Exercise

   Find the direction cosines of the sides of the triangle whose vertices are (3,5,-4), (-1,1,2) and (-5,-5,-2).

   Hint. Form each side's direction ratios by subtracting endpoints, then divide by the magnitude.

   Since each side's direction ratios come from subtracting its endpoints, and dividing by the magnitude gives the direction cosines: AB=(-4,-4,6) with magnitude sqrt68, BC=(-4,-6,-4) with magnitude sqrt68, and CA=(8,10,-2) with magnitude sqrt168.

   ✦ AB: (-4/sqrt68, -4/sqrt68, 6/sqrt68); BC: (-4/sqrt68, -6/sqrt68, -4/sqrt68); CA: (8/sqrt168, 10/sqrt168, -2/sqrt168)

Solutions written by the tuition.in editorial team and checked against lemh205.pdf (NCERT, Reprint 2026-27). Questions are referenced from the NCERT textbook for identification.