Q5. A line has direction cosines 6/7, 2/7, −3/7. What are the direction cosines of the same line taken in the opposite direction, and what are its direction angles?
−6/7, −2/7, 3/7. For the reversed direction, the angles are cos−1 (−6/7) ≈ 149.0°, cos−1 (−2/7) ≈ 106.6° and cos−1 (3/7) ≈ 64.6°. The original direction has their supplements, approximately 31.0°, 73.4° and 115.4°.
Q6. Find the direction cosines of the line joining the origin to (2, 10, −11).
The ratios are 2, 10, −11, and the distance is √(4 + 100 + 121) = 15, so the direction cosines are 2/15, 2/3, −11/15.
Q7. Find k so that A(1, 2, 3), B(2, 4, k) and C(4, 8, 11) are collinear.
AB has ratios 1, 2, k − 3 and AC has ratios 3, 6, 8. They must be proportional, with factor 3, so 3(k − 3) = 8 and k = 17/3.
Q8. Do the lines with direction ratios 2, −3, 6 and −4, 6, −12 have the same direction cosines?
Yes, up to sign. The second set is −2 times the first, so the lines are parallel, and each set of ratios gives the direction cosines ±(2/7, −3/7, 6/7): ratios fix a line’s direction only up to sign. If the lines are directed along 2, −3, 6 and along −4, 6, −12, the cosines are 2/7, −3/7, 6/7 and −2/7, 3/7, −6/7: the same line direction, taken both ways.
Q9. Beyond the NCERT text (JEE): if a line makes angles α, β, γ with the axes, show that cos 2α + cos 2β + cos 2γ = −1.
cos 2θ = 2 cos2 θ − 1, so the sum is 2(l2 + m2 + n2) − 3 = 2 − 3 = −1.
Q10. Beyond the NCERT text (JEE): a line makes angles α, β, γ, δ with the four diagonals of a cube. Show that cos2 α + cos2 β + cos2 γ + cos2 δ = 4/3.
The diagonals have direction ratios (1, 1, 1), (−1, 1, 1), (1, −1, 1) and (1, 1, −1), each of length √3. For a line with direction cosines l, m, n, the four cosines are (±l ± m ± n)/√3 with these signs. Squaring and adding, the cross terms cancel, and the sum is 4(l2 + m2 + n2)/3 = 4/3.