MathsHardClass 12

3D Geometry — Line and Plane Equations

3D Geometry

JEE Qs

30%

Hard

min

Focus on visualizing the geometry and relating vector operations (dot/cross products) to specific geometric conditions like perpendicularity, parallelism, and coplanarity.

🧮 Key Formulas

Vector equation of a line passing through a point 'a' and parallel to vector 'b': r = a + λb

Cartesian equation of a line passing through (x1, y1, z1) with direction ratios (a, b, c): (x-x1)/a = (y-y1)/b = (z-z1)/c

Vector equation of a line passing through two points 'a' and 'b': r = a + λ(b-a)

Vector equation of a plane in normal form (perpendicular distance 'p' from origin and unit normal vector n̂): r · n̂ = p

Cartesian equation of a plane in normal form: lx + my + nz = p

Vector equation of a plane passing through a point 'a' and normal to vector 'n': (r - a) · n = 0

Cartesian equation of a plane passing through (x1, y1, z1) and normal to vector (A, B, C): A(x-x1) + B(y-y1) + C(z-z1) = 0

Vector equation of a plane passing through three non-collinear points 'a', 'b', 'c': (r - a) · [(b-a) x (c-a)] = 0

Cartesian equation of a plane passing through three non-collinear points (x1,y1,z1), (x2,y2,z2), (x3,y3,z3): determinant of [[x-x1, y-y1, z-z1], [x2-x1, y2-y1, z2-z1], [x3-x1, z3-z1, z3-z1]] = 0

Intercept form of a plane: x/a + y/b + z/c = 1

Equation of a plane passing through the intersection of two planes P1=0 and P2=0: P1 + λP2 = 0

Angle between two lines r1 = a1 + λb1 and r2 = a2 + μb2: cos θ = |b1 · b2| / (|b1| |b2|)

Angle between two planes r · n1 = d1 and r · n2 = d2: cos θ = |n1 · n2| / (|n1| |n2|)

Angle between a line r = a + λb and a plane r · n = d: sin θ = |b · n| / (|b| |n|)

Perpendicular distance of a point (x1, y1, z1) from a plane Ax+By+Cz+D=0: |Ax1 + By1 + Cz1 + D| / sqrt(A^2 + B^2 + C^2)

Shortest distance between two skew lines r1 = a1 + λb1 and r2 = a2 + μb2: |(a2 - a1) · (b1 x b2)| / |b1 x b2|

Shortest distance between two parallel lines r1 = a1 + λb and r2 = a2 + μb: |(a2 - a1) x b| / |b|

✅ Key Points for JEE

1For lines, the crucial vector is the direction vector; for planes, it's the normal vector. All relative positions and angles depend on these vectors.
2Master the conversion between vector and Cartesian forms of equations for lines and planes, as problems frequently require switching between them.
3The concept of a 'family of planes' (P1 + λP2 = 0) is extremely powerful for finding a specific plane satisfying additional conditions.
4Understand the geometric significance of dot product (perpendicularity) and cross product (normal vector, area) in 3D problems.
5Pay close attention to specific cases like parallel lines/planes, perpendicular lines/planes, and coplanar lines, as these simplify formulas or imply certain conditions.

⚠️ Common Mistakes

✕Confusing the direction vector of a line with the normal vector of a plane, leading to incorrect angle calculations (e.g., using cos for line-plane angle instead of sin).
✕Sign errors in distance formulas, especially when the constant term 'D' in Ax+By+Cz+D=0 is positive/negative.
✕Applying the skew line shortest distance formula without first checking if the lines are parallel or intersecting, which require different approaches.
✕Algebraic mistakes when calculating dot products, cross products, or determinants, which are central to almost all 3D geometry problems.
✕Not properly identifying the given point and direction/normal vector from a problem description, especially when they are implicitly provided.

📝 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