MathsMediumClass 12

Direction Cosines & Direction Ratios

3D Geometry

JEE Qs

Hard

min

Master the distinction and interconversion between direction cosines and direction ratios, as their correct application is fundamental to all 3D geometry problems involving lines and planes.

🧮 Key Formulas

If a line makes angles α, β, γ with the positive x, y, z axes respectively, its direction cosines are l = cos(α), m = cos(β), n = cos(γ).

For a line passing through the origin and a point (x, y, z), its direction cosines are l = x/r, m = y/r, n = z/r, where r = sqrt(x^2 + y^2 + z^2).

The sum of the squares of direction cosines is always 1: l^2 + m^2 + n^2 = 1.

If (a, b, c) are direction ratios of a line, then its direction cosines are l = a/sqrt(a^2 + b^2 + c^2), m = b/sqrt(a^2 + b^2 + c^2), n = c/sqrt(a^2 + b^2 + c^2).

Direction ratios of a line segment joining P(x1, y1, z1) and Q(x2, y2, z2) are (x2-x1, y2-y1, z2-z1) or (x1-x2, y1-y2, z1-z2).

Direction cosines of a line segment joining P(x1, y1, z1) and Q(x2, y2, z2) are ((x2-x1)/d, (y2-y1)/d, (z2-z1)/d) where d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2).

The angle θ between two lines with direction cosines (l1, m1, n1) and (l2, m2, n2) is given by cos(θ) = |l1l2 + m1m2 + n1n2|.

The angle θ between two lines with direction ratios (a1, b1, c1) and (a2, b2, c2) is given by cos(θ) = |a1a2 + b1b2 + c1c2| / (sqrt(a1^2+b1^2+c1^2) * sqrt(a2^2+b2^2+c2^2)).

Condition for perpendicular lines: l1l2 + m1m2 + n1n2 = 0 or a1a2 + b1b2 + c1c2 = 0.

Condition for parallel lines: l1/l2 = m1/m2 = n1/n2 (if l2, m2, n2 are non-zero) or a1/a2 = b1/b2 = c1/c2 (if a2, b2, c2 are non-zero).

✅ Key Points for JEE

1Direction cosines (DCs) uniquely define the orientation of a line or vector (up to sign), whereas direction ratios (DRs) are infinite sets of numbers proportional to the DCs.
2Always remember that l^2 + m^2 + n^2 = 1. This is a fundamental identity for direction cosines and is often used to solve for an unknown direction cosine.
3When converting direction ratios (a, b, c) to direction cosines (l, m, n), the magnitude `sqrt(a^2 + b^2 + c^2)` is crucial for normalization.
4The angle between two lines can be found using either their direction cosines or direction ratios; ensure correct application of the respective formulas.
5Be mindful of the sign convention for direction cosines; a line has two sets of direction cosines (l, m, n) and (-l, -m, -n), corresponding to its two directions.

⚠️ Common Mistakes

✕Confusing direction cosines with direction ratios; DCs are normalized, DRs are not necessarily.
✕Forgetting the fundamental identity l^2 + m^2 + n^2 = 1, leading to incomplete or incorrect solutions.
✕Incorrectly calculating the denominator when converting DRs to DCs, often missing the square root or summing instead of squaring and then rooting.
✕Errors in applying the dot product formula for the angle between two lines when using DRs, particularly forgetting the denominators for normalization.

📝 Practice Questions

See all

Q4. Let P be the foot of the perpendicular from the point (1, 2, 2) on the line L : x−11 = y+1−1 = z−22 . Let the line →r = (−^i + ^j −2^k) + λ(^i −^j + ^k), λ ∈R, intersect the line L at Q . Then 2(PQ)2 is equal to : (1) 25 (2) 19 (3) 29 (4) 27

2025·MCQMedium

Q16.Let a straight line L pass through the point P(2, −1, 3) and be perpendicular to the lines x−12 = y+11 = z−3−2 and x−3 1 = y−23 = z+24 . If the line L intersects the yz -plane at the point Q , then the distance between the points P and Q is : (1) √10 (2) 2√3 (3) 2 (4) 3

2025·MCQMedium

Q8. Let L1 : x−12 = y−23 = z−34 and L2 : x−23 = y−44 = z−55 be two lines. Then which of the following points lies on the line of the shortest distance between L1 and L2 ? (1) ( 143 , −3, 223 ) (2) (−53 , −7, 1) (3) (2, 3, 13 ) (4) ( 83 , −1, 13 )

2025·MCQHard

Q25.Let L1 : x−13 = y−1−1 = z+10 and L2 : x−22 = 0y = z+4α , α ∈R, be two lines, which intersect at the point B. If P is the foot of perpendicular from the point A(1, 1, −1) on L2 , then the value of 26α( PB)2 is _________

2025·NumericalMedium

Q14.The perpendicular distance, of the line x−1 2 = −1 = z+32 from the point P(2, −10, 1), is : (1) 6 (2) 5√2 (3) 4√3 (4) 3√5

2025·MCQMedium

Q3. Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be ^i + 2^j + ^k,^i + 3^j −2^k and 2^i + ^j −^k respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through A of the triangle ABC at the point E . If the length of AD is √110 and the volume of the 3 tetrahedron is √805 , then the position vector of E is 6√2 (1) 12 1 (7^i + 4^j + 3^k) (2) 12 (^i + 4^j + 7^k) (3) 1 6 (12^i + 12^j + ^k) (4) 16 (7^i + 12^j + ^k)

2025·Multi conceptHard

NCERT Chapters

Class 12 Mathematics Ch 11: Three Dimensional Geometry