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MathsMediumClass 12

Position Vector — Midpoint, section formula

Vectors

12

JEE Qs

8%

Hard

60

min

Master the derivation of the section formula to confidently apply it in various geometric problems and avoid common sign errors, especially in external division.

🧮 Key Formulas

Position vector of point A from origin O: vec(OA) = vec(a)
Position vector of a point R dividing line segment AB internally in the ratio m:n: vec(r) = (n*vec(a) + m*vec(b)) / (m + n)
Position vector of a point R dividing line segment AB externally in the ratio m:n: vec(r) = (n*vec(a) - m*vec(b)) / (n - m)
Midpoint formula (special case of internal division 1:1): vec(r) = (vec(a) + vec(b)) / 2
Centroid of a triangle ABC with position vectors vec(a), vec(b), vec(c): vec(G) = (vec(a) + vec(b) + vec(c)) / 3
Centroid of a tetrahedron ABCD with position vectors vec(a), vec(b), vec(c), vec(d): vec(G) = (vec(a) + vec(b) + vec(c) + vec(d)) / 4

✅ Key Points for JEE

  • 1The section formula is a vector-based weighted average of the position vectors of the endpoints, with weights inversely proportional to the segment lengths.
  • 2External division can be conceptualized as internal division with one ratio being negative (e.g., m:(-n) or (-m):n), which simplifies derivation recall.
  • 3Always ensure consistency in assigning 'm' and 'n' to the respective position vectors. If R divides AB in ratio m:n, then 'm' is associated with B (vec(b)) and 'n' with A (vec(a)).
  • 4Position vectors are always measured relative to a fixed origin, typically denoted as O.
  • 5The section formula is a powerful tool for proving collinearity (if the dividing point lies on the line segment) and concurrency (if multiple line segments intersect at a common point satisfying the section formula).

⚠️ Common Mistakes

  • ✕Swapping 'm' and 'n' or associating them with the incorrect position vector (i.e., using m*vec(a) + n*vec(b) instead of n*vec(a) + m*vec(b)).
  • ✕Confusing the sign for internal vs. external division, leading to incorrect calculations.
  • ✕Incorrectly identifying the ratio m:n, especially in problems where the dividing point is given in relation to the entire line or one segment.
  • ✕Forgetting that if a point lies on the line AB extended, it implies external division, and misapplying the internal division formula.

📝 Practice Questions

See all

Q10.Let →a and →b be two unit vectors such that the angle between them is . If λ→a + 2→b and 3→a −λ→b are 3 perpendicular to each other, then the number of values of λ in [−1, 3] is : (1) 2 (2) 1 (3) 0 (4) 3 e 1 x x loge α

2025·MCQEasy

Q10.Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC , divides the arc −−−length of arc AB 1 → → → AC such that length of arc BC = 5 , and OC = αOA + βOB , then α + √2(√3 −1)β is equal to (1) 2√3 (2) 2 −√3 (3) 5√3 (4) 2 + √3 f ∘g is

2025·Multi conceptHard

Q23.Let →c be the projection vector of →b = λ^i + 4^k, λ > 0, on the vector →a = ^i + 2^j + 2^k. If |→a + →c| = 7, then the area of the parallelogram formed by the vectors →b and →c is ________

2025·NumericalMedium

Q4. Let a line pass through two distinct points P(−2, −1, 3) and Q , and be parallel to the vector 3^i + 2^j + 2^k. If the distance of the point Q from the point R(1, 3, 3) is 5 , then the square of the area of △PQR is equal to : (1) 148 (2) 136 (3) 144 (4) 140

2025·MCQMedium

Q12.Let →a = 3^i −^j + 2^k, b =→a× (^i −2^k) and→c= b × ^k. Then the projection of→c−2^j on →a is : (1) 2√14 (2) √14 (3) 3√7 (4) 2√7

2025·MCQMedium

Q48.Two particles are located at equal distance from origin. The position vectors of those are represented by ¯A = 2^i + 3n^j + 2^k and ¯B = 2^i −2^j + 4p^k, respectively. If both the vectors are at right angle to each other, the value of n−1 is _____ .

2025·NumericalEasy

NCERT Chapters

  • Class 12 Mathematics Ch 10: Vector Algebra