Q5. Find the angle between the lines with direction cosines 1/√2, 1/2, 1/2 and −1/√2, 1/2, 1/2.
cos θ = |−1/2 + 1/4 + 1/4| = 0, so the lines are perpendicular.
Q6. Find the acute angle between two diagonals of a cube.
Take the cube with edges along the axes. Two diagonals have direction ratios 1, 1, 1 and 1, 1, −1, so cos θ = |1 + 1 − 1|/(√3 × √3) = 1/3, and θ = cos−1 (1/3) ≈ 70.5°.
Q7. Find k if the lines (x + 1)/(k + 1) = (y − 2)/4 = z/2 and x/3 = y/6 = (z − 1)/3 are parallel.
The ratios must be proportional: 4/6 = 2/3, so (k + 1)/3 = 2/3 and k = 1.
Q8. Find sin θ for the lines of Example 2, and check it against cos θ.
Using NCERT’s sin θ formula: ab − ab = 2 − 8 = −6, bc − bc = 16 − 1 = 15, ca − ca = 4 − 16 = −12. So sin θ = √(36 + 225 + 144)/27 = √405/27 = √5/3, and sin2 θ + cos2 θ = 5/9 + 4/9 = 1.
Q9. A line makes 60° with the x-axis and 90° with the y-axis. Find the angle it makes with the line x = y = z.
Its third direction cosine satisfies n2 = 1 − 1/4 − 0 = 3/4, so the line has direction cosines 1/2, 0, ±√3/2. With 1/√3, 1/√3, 1/√3, cos θ = |1/2 ± √3/2|/√3, that is, (1 + √3)/(2√3) ≈ 0.789 or (√3 − 1)/(2√3) ≈ 0.211: θ ≈ 37.9° or about 77.8°, one for each choice of n.
Q10. Beyond the NCERT text (JEE): find the angle between the lines whose direction cosines satisfy l + m + n = 0 and l2 + m2 − n2 = 0.
From the first, n = −(l + m); then l2 + m2 = (l + m)2 gives lm = 0. If l = 0, then m = −n: direction 0, 1, −1. If m = 0, then l = −n: direction 1, 0, −1. cos θ = |0 + 0 + 1|/(√2 × √2) = 1/2, so θ = 60°.
Q11. Find k if the lines (x − 1)/(−3) = (y − 2)/(2k) = (z − 3)/2 and (x − 1)/(3k) = (y − 1)/1 = (z − 6)/(−5) are perpendicular (NCERT Miscellaneous Exercise on Chapter 11, Q3, page 391).
The direction ratios are −3, 2k, 2 and 3k, 1, −5, so −9k + 2k − 10 = 0, and k = −10/7.