Miscellaneous Exercise — Three Dimensional Geometry

1. 13 marksNCERT Exercise

   Find the angle between the lines whose direction ratios are a, b, c and b-c, c-a, a-b.

   Hint. Compute the dot product of the two sets of direction ratios and simplify.

   Since the dot product is a(b-c)+b(c-a)+c(a-b) = ab-ac+bc-ab+ac-bc = 0, which vanishes identically for all a, b, c, the two lines are always perpendicular.

   ✦ 90 degrees (the lines are always perpendicular)

2. 23 marksNCERT Exercise

   Find the equation of a line parallel to the x-axis and passing through the origin.

   Hint. A line parallel to the x-axis has direction ratios (1,0,0), and it passes through the origin.

   Since a line parallel to the x-axis has direction vector i (direction ratios 1,0,0), and it passes through the origin whose position vector is 0, substituting into r = a + lambda.b gives the equation directly.

   ✦ r = lambda.i, i.e. x/1 = y/0 = z/0

3. 33 marksNCERT Exercise

   If the lines (x-1)/(-3)=(y-2)/(2k)=(z-3)/2 and (x-1)/(3k)=(y-1)/1=(z-6)/(-5) are perpendicular, find the value of k.

   Hint. Set the dot product of the two sets of direction ratios equal to zero and solve for k.

   Since the direction ratios are (-3, 2k, 2) and (3k, 1, -5), setting their dot product to zero gives -9k + 2k - 10 = 0, so -7k = 10.

   ✦ k = -10/7

4. 43 marksNCERT Exercise

   Find the shortest distance between the lines r=(6i+2j+2k)+lambda(i-2j+2k) and r=(-4i-k)+mu(3i-2j-2k).

   Hint. Apply the skew-lines shortest-distance formula with the given points and direction vectors.

   With a1=(6,2,2), b1=(1,-2,2), a2=(-4,0,-1), b2=(3,-2,-2), computing b1 x b2 and applying |(b1 x b2).(a2-a1)|/|b1 x b2| gives the shortest distance.

   ✦ 9 units

5. 53 marksNCERT Exercise

   Find the vector equation of the line passing through the point (1,2,-4) and perpendicular to the two lines (x-8)/3=(y+19)/(-16)=(z-10)/7 and (x-15)/3=(y-29)/8=(z-5)/(-5).

   Hint. A vector perpendicular to both lines is the cross product of their direction vectors.

   Since the two direction vectors are (3,-16,7) and (3,8,-5), their cross product is (24,36,72), which simplifies to the direction ratios (2,3,6); because the line passes through (1,2,-4), substituting into r = a + lambda.b gives the vector equation.

   ✦ r = (i+2j-4k) + lambda(2i+3j+6k)

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.