Why this chapter is procedural
Ask for the distance from a point to a plane and there is exactly one formula. Ask for the angle between two planes and there is exactly one formula. Ask for the image of a point in a plane and there is one three-step procedure - drop a perpendicular, find the foot, extend by the same distance. Nothing requires invention.
The consequence is that this chapter rewards drilling the standard types until each is automatic, more than it rewards thinking hard about any one problem. It is also why it pairs so naturally with Vector Algebra - the operations are the dot and cross products, dressed in geometric language.
How JEE actually asks 3D Geometry
NTA publishes no chapter-wise weightage, so figures circulating online are coaching estimates from past papers; check the official NTA information bulletin for your session's syllabus and pattern.
The recurring types are: find the equation of a line or plane from given conditions, find the angle between two lines, two planes, or a line and a plane, find a distance from a point to a plane or between skew lines, and find the foot of a perpendicular or the image of a point. Coplanarity of two lines rounds it out.
Key concepts, compressed
- ✓Direction cosines are the cosines of the angles a line makes with the axes, and they are the components of a unit vector along it.
- ✓Direction ratios are any triple proportional to the direction cosines, so they are determined only up to a scale factor.
- ✓A line is fixed by a point on it and a direction; a plane is fixed by a point on it and a normal direction.
- ✓The angle between two lines or two planes comes from the dot product of their directions or normals.
- ✓A line and a plane are perpendicular when the line's direction is parallel to the normal, and parallel when it is perpendicular to the normal - the reverse of what intuition suggests.
- ✓Skew lines neither meet nor are parallel, and their shortest distance is measured along the common perpendicular.
Formulas you need before attempting the questions
| Quantity | Formula | Note |
|---|---|---|
| Direction cosines | l² + m² + n² = 1 | |
| From direction ratios | divide each by √(a² + b² + c²) | |
| Line, vector form | r = a + λb | point a, direction b |
| Line, Cartesian form | (x − x₁)/a = (y − y₁)/b = (z − z₁)/c | |
| Plane, vector form | r·n = d | |
| Plane, Cartesian form | ax + by + cz + d = 0 | normal is (a, b, c) |
| Angle between two lines | cos θ = |b₁·b₂|/(|b₁||b₂|) | |
| Angle between two planes | cos θ = |n₁·n₂|/(|n₁||n₂|) | |
| Angle between line and plane | sin θ = |b·n|/(|b||n|) | sine, not cosine |
| Point to plane distance | |ax₁ + by₁ + cz₁ + d| / √(a² + b² + c²) | |
| Skew lines, shortest distance | |(a₂ − a₁)·(b₁ × b₂)| / |b₁ × b₂| | |
| Perpendicular lines | a₁a₂ + b₁b₂ + c₁c₂ = 0 | |
| Perpendicular planes | n₁·n₂ = 0 |
The five mistakes that cost the most marks
- ✓Using cosine for the angle between a line and a plane. That angle is measured from the plane, not from the normal, so it uses sine.
- ✓Treating the coefficients of a plane as a point on it. In ax + by + cz + d = 0 the triple (a, b, c) is the normal vector.
- ✓Assuming two non-intersecting lines must be parallel. In three dimensions they can be skew, which is a genuinely new possibility.
- ✓Dropping the absolute value in a distance formula. The sign only indicates which side of the plane the point lies on.
- ✓Applying Kepler-style intuition from two dimensions - for instance assuming two planes must intersect in a point. Two planes meet in a line, and three are needed for a point.
Practice set 1: direction cosines and ratios
1. What relation do direction cosines satisfy?
l² + m² + n² = 1. They are the components of a unit vector along the line, and a unit vector has magnitude one. This is the fastest check on any set of direction cosines - if the squares do not sum to one, they are direction ratios that have not been normalised.
2. Find the direction ratios of the line through (1, 2, 3) and (4, 6, 3).
(3, 4, 0). Subtract the coordinates: (4 − 1, 6 − 2, 3 − 3) = (3, 4, 0). The zero third component means the line is parallel to the xy-plane, since its z-coordinate never changes. Direction ratios may be scaled freely, so (6, 8, 0) describes the same direction.
3. Find the direction cosines of that line.
(3/5, 4/5, 0). Divide the direction ratios by their magnitude √(9 + 16 + 0) = 5. Check: (3/5)² + (4/5)² + 0² = 9/25 + 16/25 = 1, as required. The zero remains zero, confirming the line makes a right angle with the z-axis.
4. Are direction ratios unique?
No. Any non-zero scalar multiple of a valid set is equally valid, since only the direction matters. Direction cosines, by contrast, are unique up to an overall sign - reversing all three describes the same line traversed in the opposite sense.
5. How do you find the angle between two lines given their direction ratios?
cos θ = |b₁·b₂|/(|b₁||b₂|), the dot product of the direction vectors over the product of their magnitudes. The absolute value is taken because a line has no preferred direction, so the acute angle is reported. Without it you could get an obtuse answer for the same pair of lines.
6. What is the condition for two lines to be perpendicular?
a₁a₂ + b₁b₂ + c₁c₂ = 0, the dot product of the direction ratios vanishing. For parallel lines the condition is instead that the ratios are proportional: a₁/a₂ = b₁/b₂ = c₁/c₂. These are the same perpendicular and parallel tests from Vector Algebra in coordinate form.
Practice this now
Practice set 2: equations of lines
7. Write the vector equation of a line.
r = a + λb, where a is the position vector of a known point on the line, b is a direction vector, and λ is a parameter running over the reals. Every point of the line is obtained by some λ. Two different values of a describing the same line are equally valid, which is why line equations are not unique.
8. Write the Cartesian form of a line through (x₁, y₁, z₁) with direction ratios (a, b, c).
(x − x₁)/a = (y − y₁)/b = (z − z₁)/c. Each fraction equals the parameter λ, which is how the vector and Cartesian forms correspond. If one of the direction ratios is zero the corresponding equation is written separately, for instance z = z₁, since division by zero is not permitted.
9. How do you find the equation of the line through two points?
Take one point as a and the difference of the two position vectors as the direction b, then write r = a + λb. This is exactly the calculation in question 2 followed by question 7. Either point may serve as a, giving different but equivalent equations.
10. How do you check whether a point lies on a given line?
Substitute its coordinates into the Cartesian form and check that all three fractions give the same value of the parameter. If they disagree the point is off the line. Checking only one or two of the fractions is a common shortcut that fails - all three must agree.
11. When are two lines parallel?
When their direction ratios are proportional, so their direction vectors are scalar multiples of each other. Parallel lines may be distinct or identical, and to distinguish those you check whether a point of one lies on the other. Distinct parallel lines never meet but are not skew.
12. What are skew lines?
Lines that neither intersect nor are parallel. They are possible only in three or more dimensions, because in a plane two non-parallel lines must cross. Skew lines lie in parallel planes, and the shortest distance between them is measured along their common perpendicular - the subject of question 24.
Practice set 3: equations of planes
13. Write the vector equation of a plane.
r·n = d, where n is a normal vector to the plane. Every position vector r in the plane has the same projection onto the normal, which is what makes the dot product constant. Dividing through by |n| gives the normal form, in which d becomes the perpendicular distance from the origin.
14. In the plane ax + by + cz + d = 0, what does (a, b, c) represent?
The normal vector to the plane - a direction perpendicular to it, not a point on it. This is the single most useful fact in the chapter, since almost every plane question begins by extracting the normal. Treating (a, b, c) as a point is a persistent error.
15. Find the equation of the plane through (1, 2, 3) with normal (2, −1, 4).
2x − y + 4z = 12. The plane has the form 2x − y + 4z = d, and substituting the given point gives 2(1) − 2 + 4(3) = 2 − 2 + 12 = 12. Equivalently 2x − y + 4z − 12 = 0. This two-step method - write the form from the normal, then fix the constant from the point - handles every such question.
16. How do you find the plane through three given points?
Form two vectors by subtracting one point from the other two, take their cross product to obtain a normal, then use that normal with any one of the points as in question 15. If the cross product comes out as the zero vector, the three points are collinear and no unique plane exists.
17. What is the intercept form of a plane?
x/a + y/b + z/c = 1, where a, b and c are the intercepts on the three axes. Setting y = z = 0 gives x = a, confirming the interpretation. This form is convenient when a question specifies intercepts directly, and it fails only for planes through the origin, where no non-zero intercepts exist.
18. When are two planes parallel?
When their normal vectors are proportional, so the planes have the same orientation. For example 2x − y + z = 3 and 4x − 2y + 2z = 7 are parallel, since their normals (2, −1, 1) and (4, −2, 2) are proportional but the constants are not in the same ratio - which is what makes them distinct rather than identical.
19. When are two planes perpendicular?
When their normals are perpendicular, that is n₁·n₂ = 0. Note the pattern: relations between planes reduce to relations between their normals, and relations between lines reduce to relations between their directions. Only the line-and-plane case inverts, which is the subject of the next set.
Practice this now
Practice set 4: angles and distances
20. How do you find the angle between two planes?
cos θ = |n₁·n₂|/(|n₁||n₂|), using the two normal vectors. The angle between the planes equals the angle between their normals, and the absolute value ensures the acute angle is reported. This is the same formula as for two lines, with normals substituted for directions.
21. How do you find the angle between a line and a plane?
sin θ = |b·n|/(|b||n|), using sine rather than cosine. The formula computes the angle between the line and the normal, and the angle with the plane is its complement - which is exactly what replacing cosine with sine achieves. Using cosine here is the most common error in the chapter.
22. Find the distance from the point (1, 2, 3) to the plane 2x + 3y + z − 6 = 0.
5/√14, about 1.34. Substitute into |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²): the numerator is |2 + 6 + 3 − 6| = 5 and the denominator is √(4 + 9 + 1) = √14. The absolute value matters - without it the sign would only tell you which side of the plane the point is on.
23. How do you find the distance between two parallel planes?
Make the coefficients identical in both equations, then the distance is the difference of the constants divided by the magnitude of the normal. Equivalently, take any point on one plane and compute its distance to the other using the point-to-plane formula, which is the safer route if the scaling is awkward.
24. What is the shortest distance between two skew lines?
|(a₂ − a₁)·(b₁ × b₂)| / |b₁ × b₂|. The cross product b₁ × b₂ gives the direction of the common perpendicular, and projecting the vector joining the two lines onto that direction gives the distance. If b₁ × b₂ is zero the lines are parallel, not skew, and a different formula applies.
25. How do you find the foot of the perpendicular from a point to a plane?
Write the line through the point in the direction of the plane's normal, then find where it meets the plane by substituting the parametric coordinates into the plane equation and solving for the parameter. Substituting that value back gives the foot. This three-step procedure works every time.
26. How do you find the image of a point in a plane?
Find the foot of the perpendicular as in question 25, then continue the same distance beyond it - the foot is the midpoint of the point and its image. Practically, if the parameter at the foot is λ, the image is obtained by using 2λ instead. Forgetting to double is the standard error.
Practice set 5: mixed
27. What is the condition for a line to lie in a plane?
Two conditions together: the line's direction must be perpendicular to the plane's normal, so b·n = 0, and one point of the line must satisfy the plane's equation. The first alone only makes the line parallel to the plane; the second pins it into the plane. Checking only one is incomplete.
28. What is the condition for two lines to be coplanar?
That the scalar triple product of the vector joining a point on each line with the two direction vectors is zero: [(a₂ − a₁) b₁ b₂] = 0. This is exactly the numerator of the skew-line distance formula, so coplanar lines are those whose shortest distance is zero - they either intersect or are parallel.
29. In what do two non-parallel planes intersect?
A line, never a point. Two planes constrain two of the three degrees of freedom, leaving one free - which is a line. A third non-parallel plane is required to pin down a point, which is the geometric meaning of solving three linear equations in three unknowns.
30. How do you find a vector normal to a plane containing two given directions?
Take their cross product, which by construction is perpendicular to both and therefore normal to the plane they span. This is the bridge between Vector Algebra and this chapter, and it is the first step in finding the plane through three points or the common perpendicular to two skew lines.
How to study this chapter efficiently
- ✓Drill the six standard types until each is automatic: line equation, plane equation, angle, distance, foot of perpendicular, image of a point.
- ✓Write down the normal vector as the very first step of any plane question. Most of the chapter follows from it.
- ✓Keep the sine and cosine cases separate: cosine for line-to-line and plane-to-plane, sine for line-to-plane.
- ✓Check any set of direction cosines by confirming the squares sum to one.
- ✓For skew lines, compute b₁ × b₂ first. If it is zero the lines are parallel and the skew formula does not apply.
- ✓Prepare this chapter with Vector Algebra rather than after it - the operations are identical and only the vocabulary differs.
Turn this into active practice
Because the procedures are fixed, the failure mode here is not confusion but misclassification - reaching for the plane-to-plane formula on a line-and-plane question, or the parallel-line distance on a skew pair. That only shows up when the question type is not announced.
The JEE 3D Geometry quiz on QUFF generates fresh questions across direction cosines, lines, planes, angles and distances, marks them instantly and explains each answer. Do a mixed set and, for each error, name which of the six standard types the question actually was. Those misclassifications repeat, and naming them is the fix.