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The section formula looks easy, and that is why it costs marks. Students swap the ratio, forget the minus sign for external division, or cannot work backwards to find a ratio. This post explains where the formula now sits in your books, derives it in one line, and works through the main question types, with a check for each.
Is the section formula still in the syllabus?
It depends on which document you read. The Syllabus for JEE (Main) 2026 lists "section formula" in Unit 11, Three Dimensional Geometry, along with distance, direction ratios and lines.
The NCERT Class 11 chapter, Introduction to Three Dimensional Geometry (2026-27 reprint), now stops after the distance formula: its sections end at 11.4, Distance between Two Points. The section formula was removed in the rationalised edition. The CBSE Mathematics curriculum for 2026-27 still names "Section formula" under Class 11 three-dimensional geometry, but in the list of topics assessed only formatively, with NCERT material enclosed for reference.
The vector form is still in the book. The Class 12 chapter Vector Algebra has section 10.5.3, Section formula, covering internal and external division. So JEE Main aspirants should learn it from the Class 12 chapter and practise it in coordinates. Check the syllabus for your exam year.
Where does the formula come from?
Let R divide AB internally so that AR : RB = m : n. Then R is m/(m + n) of the way from A to B. In position vectors:
r = a + (m/(m + n))(b − a) = (mb + na)/(m + n)
Read off each coordinate and you get R = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n)), where A = (x₁, y₁, z₁) and B = (x₂, y₂, z₂).
For external division, R lies on the line AB extended, with AR : RB = m : n measured as lengths. The formula becomes (mx₂ − nx₁)/(m − n), and the same for y and z. The midpoint is the case m = n.
Worked example 1: internal and external division
Take A(2, −3, 4) and B(8, 0, 10).
Internally in 1 : 2. x = (1 × 8 + 2 × 2)/3 = 12/3 = 4. y = (1 × 0 + 2 × (−3))/3 = −2. z = (1 × 10 + 2 × 4)/3 = 6. So R = (4, −2, 6).
Check: B − A = (6, 3, 6), which has length 9. R − A = (2, 1, 2), length 3, and B − R = (4, 2, 4), length 6. The ratio 3 : 6 is 1 : 2.
Externally in 1 : 2. x = (1 × 8 − 2 × 2)/(1 − 2) = 4/(−1) = −4. y = (0 + 6)/(−1) = −6. z = (10 − 8)/(−1) = −2. So R = (−4, −6, −2).
Check: R − A = (−6, −3, −6) = −(B − A). R is 9 units from A on the side away from B, and 18 units from B, which gives 1 : 2 again. When m < n, an external point lies beyond A; when m > n, beyond B.
Trisection points. The two points that cut AB into three equal parts divide it in 1 : 2 and 2 : 1. The first is (4, −2, 6) from above. The second is ((2 × 8 + 2)/3, (0 − 3)/3, (20 + 4)/3) = (6, −1, 8).
Worked example 2: which ratio does a coordinate plane cut?
Find the ratio in which the yz-plane divides the join of P(−2, 4, 7) and Q(3, −5, 8), and the point of division.
Let the ratio be k : 1. The point is ((3k − 2)/(k + 1), (−5k + 4)/(k + 1), (8k + 7)/(k + 1)). On the yz-plane, x = 0, so 3k − 2 = 0 and k = 2/3. The ratio is 2 : 3, internal because k is positive.
The point: y = (−10/3 + 4)/(5/3) = (2/3)/(5/3) = 2/5, and z = (16/3 + 7)/(5/3) = 37/5. So the plane meets PQ at (0, 2/5, 37/5).
Now try the xy-plane on the same segment. Setting z = 0 gives 8k + 7 = 0, so k = −7/8. A negative k means external division in 7 : 8. That makes sense: both P and Q have positive z, so they lie on the same side of the xy-plane and the segment never crosses it. The extended line meets the plane at (−37, 67, 0).
The k : 1 method uses one unknown, and the sign of k tells you at once whether the division is internal or external.
Worked example 3: using the formula as a collinearity test
Do A(2, −3, 4), B(8, 0, 10) and C(6, −1, 8) lie on one line? Suppose C divides AB in k : 1. From x: (8k + 2)/(k + 1) = 6 gives k = 2. Check y: (0 + (−3))/3 = −1. Check z: (20 + 4)/3 = 8. All three agree, so the points are collinear and C divides AB in 2 : 1. If one coordinate had given a different k, the points would not be collinear. Our post on the position vector test for collinear points does the same job with vectors.
Which mistakes cost the most marks?
Swapping m and n. Writing (mx₁ + nx₂)/(m + n) puts the point near the wrong end. Check with a distance.
Dropping the minus sign in external division. If your "external" point lies between A and B, the sign is wrong.
Expecting a positive ratio every time. A negative k is information, not an error.
Finding k from one coordinate and stopping. In collinearity questions, every coordinate must give the same k.
A practice routine for one week
Day 1: Derive the internal and external formulas from r = a + t(b − a) without looking.
Day 2: Five internal and five external divisions, each checked with two distances.
Day 3: Five "which ratio does this plane cut?" questions using k : 1. Predict internal or external from the signs before solving.
Day 4: Five collinearity tests by section formula, then the same five by vectors.
Day 5: A mixed set from 3D geometry and vectors, timed.
If you want to see a point slide along a segment as the ratio changes, Fermi's Simulation Lab has interactive simulations for ideas that are easier to see than to read. For the line equations that come next, read our guide to 3D geometry for JEE.
Tonight, find where the yz-plane, the xz-plane and the xy-plane each cut the join of (1, −2, 3) and (−3, 4, −5), and say whether each division is internal or external.
