Q5. Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are ±(1/√3, 1/√3, 1/√3) (NCERT Miscellaneous Exercise on Chapter 10, Q11, page 372).
Equal angles give l = m = n. Then l2 + m2 + n2 = 3l2 = 1, so l = ±1/√3, and all three cosines are 1/√3 or all are −1/√3.
Q6. Can 1/2, 1/2, 1/2 be the direction cosines of a vector? Can they be its direction ratios?
Not cosines: their squares add up to 3/4, not 1. They can be direction ratios, of the vector with direction cosines 1/√3, 1/√3, 1/√3.
Q7. A point P is at distance 6 from the origin, and OP has direction cosines 1/3, 2/3, 2/3. Find P.
P is (lr, mr, nr) with r = 6, that is, (2, 4, 4). Check: √(4 + 16 + 16) = 6.
Q8. Find the direction cosines of a vector with direction ratios 3, −6, 2.
√(9 + 36 + 4) = 7, so the direction cosines are 3/7, −6/7 and 2/7 when the vector points in the direction (3, −6, 2); their simultaneous negatives describe the opposite orientation if only the line is specified.
Q9. Beyond the NCERT text (JEE): if a vector makes angles α, β, γ with the axes, show that sin2 α + sin2 β + sin2 γ = 2.
Each sin2 is 1 − cos2, so the sum is 3 − (l2 + m2 + n2) = 3 − 1 = 2.
Q10. A vector is perpendicular to the z-axis and makes equal acute angles with the x- and y-axes. Find its direction cosines and these angles.
γ = 90°, so n = 0, and l = m with 2l2 = 1. For acute angles l = m = 1/√2, so the direction cosines are 1/√2, 1/√2, 0, and α = β = 45°.