MathsMediumClass 12

Angle Between Line and Plane

3D Geometry

JEE Qs

Hard

min

Consistently identify the direction vector of the line and the normal vector of the plane, then apply the `sin(θ)` formula correctly with absolute values to find the acute angle.

🧮 Key Formulas

Line: r = a + λb (direction vector b)

Plane: r . n = d or Ax + By + Cz + D = 0 (normal vector n = Ai + Bj + Ck)

If θ is the angle between the line and the plane, then sin(θ) = |b . n| / (|b| |n|)

In Cartesian form (line direction ratios l, m, n; plane normal direction ratios A, B, C): sin(θ) = |Al + Bm + Cn| / (sqrt(A^2 + B^2 + C^2) * sqrt(l^2 + m^2 + n^2))

✅ Key Points for JEE

1The angle between a line and a plane is defined as the angle between the line and its projection on the plane.
2The core formula calculates the sine of the required angle, not the cosine. This is because the dot product directly yields the angle between the line's direction vector and the plane's normal vector, which is complementary to the angle with the plane.
3Always identify the direction vector of the line (`b`) and the normal vector of the plane (`n`) correctly from their given equations.
4Use the absolute value of the dot product to ensure you find the acute angle between the line and the plane.

⚠️ Common Mistakes

✕Using `cos(θ)` instead of `sin(θ)` in the formula, which incorrectly gives the angle between the line and the plane's normal.
✕Incorrectly extracting the direction vector of the line or the normal vector of the plane from their equations (e.g., confusing coordinates of a point on the line with its direction ratios).
✕Forgetting to take the absolute value of the dot product, which might lead to an obtuse angle while the acute angle is typically expected.

📝 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 Maths Ch 10: Vector Algebra
Class 12 Maths Ch 11: Three Dimensional Geometry

🚀 This Unlocks

Family of Planes