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CBSE · Class 12 · Mathematics

Vector Joining Two Points and the Section Formula (Ch 10: 10.5.1–10.5.3)

2 concepts · 16 practice questions with answers · Free to read

Vector joining two points

From daily life

To tell a friend how to walk from your gate to the school, you say how far east, how far north and how far up. This triple of numbers is the vector between the two places.

Simple definition

The vector from the point P(x₁, y₁, z₁) to Q(x₂, y₂, z₂) is found by subtracting the coordinates of the start from those of the end. Its magnitude is the distance between the two points.

Formula: PQ = (x₂ − x₁)i + (y₂ − y₁)j + (z₂ − z₁)k, |PQ| = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)

Rule 1: Terminal minus initial — Subtract in this order.

  • Easy: For P(1, 2, 3) and Q(3, 4, 7), PQ = 2i + 2j + 4k and |PQ| = 2√6.
  • Medium: The vector from (2, 3, 1) to (5, 7, 13) is 3i + 4j + 12k, with magnitude 13.
  • Exam-type: If AB = 4i + 3j and A = (1, 1), then B = (5, 4).

Rule 2: Collinear points — Two vectors along the same line are proportional.

  • Easy: For A(1, 2, 3), B(3, 4, 5) and C(5, 6, 7), AB = BC = 2i + 2j + 2k, so A, B, C are collinear.
  • Medium: The position vector of P(−1, 2, 4) has magnitude √21.
  • Exam-type: The vector from (1, 1, 1) to (4, 5, x) has magnitude 13 when (x − 1)² = 144, so x = 13 or −11.
Remember

PQ = Q − P. The reverse vector QP is the negative of PQ.

Common mistake

Subtracting the end from the start, which gives the opposite direction.

Memory clue

End minus start.

Practice questions

Easy
  1. 1Find the vector PQ for P(1, 2, 3) and Q(3, 4, 7).
  2. 2Find the magnitude of PQ for P(1, 2, 3) and Q(3, 4, 7).
  3. 3Find the magnitude of the position vector of P(−1, 2, 4).
Medium
  1. 4Write the components of the vector 3i − 2j + 6k and find its magnitude.
  2. 5Show that A(1, 2, 3), B(3, 4, 5) and C(5, 6, 7) are collinear.
  3. 6If AB = 4i + 3j and A is (1, 1), find B.
Challenging
  1. 7Find the vector from (2, 3, 1) to (5, 7, 13) and its magnitude.
  2. 8Find x so that the vector from (1, 1, 1) to (4, 5, x) has magnitude 13.
Show answers (try them yourself first)
  1. 12i + 2j + 4k
  2. 2√(4 + 4 + 16) = 2√6
  3. 3√(1 + 4 + 16) = √21
  4. 4Components 3, −2 and 6, and the magnitude is 7
  5. 5AB = 2i + 2j + 2k = BC, so they are collinear
  6. 6B = (1 + 4, 1 + 3) = (5, 4)
  7. 73i + 4j + 12k, magnitude 13
  8. 89 + 16 + (x − 1)² = 169 gives (x − 1)² = 144, so x = 13 or x = −11

Section formula

From daily life

A shopkeeper places a price tag between two shelves, 1 part from one and 2 parts from the other. The section formula finds the position of such a dividing point.

Simple definition

A point R divides the segment from A to B in the ratio m : n internally if R lies between A and B, and externally if R lies on the extension. The midpoint is the case m = n for internal division.

Formula: Internal: R = (mb + na)/(m + n). External: R = (mb − na)/(m − n). Midpoint: (a + b)/2

Rule 1: Internal division — The coordinates are weighted by the ratio.

  • Easy: The midpoint of A(2, 4, 6) and B(4, 8, 10) is (3, 6, 8).
  • Medium: The point dividing A(1, 2, 3) and B(7, 8, 9) in the ratio 1 : 2 internally is (3, 4, 5).
  • Exam-type: If M(2, 3, 4) is the midpoint of P(1, 1, 1) and Q, then Q = (3, 5, 7).

Rule 2: External division and centroid — Subtract instead of adding.

  • Easy: The point dividing A(1, 1, 1) and B(3, 3, 3) externally in 3 : 1 is (4, 4, 4).
  • Medium: The centroid of the triangle with vertices (1, 2, 3), (4, 5, 6) and (7, 8, 9) is (4, 5, 6).
  • Exam-type: The point (3, 4, 5) divides A(1, 2, 3) and B(7, 8, 9) in the ratio (3 − 1) : (7 − 3) = 1 : 2.
Remember

In m : n the number m goes with the second point B, and n goes with the first point A.

Common mistake

Putting m with the first point A.

Memory clue

Cross-weights: m with B, n with A.

Practice questions

Easy
  1. 1Find the midpoint of A(2, 4, 6) and B(4, 8, 10).
  2. 2Find the point dividing A(1, 2, 3) and B(7, 8, 9) in the ratio 1 : 2 internally.
  3. 3Find the point dividing A(0, 0, 0) and B(5, 10, 15) in the ratio 2 : 3 internally.
Medium
  1. 4Find the point dividing A(1, 1, 1) and B(3, 3, 3) in the ratio 3 : 1 externally.
  2. 5Find the centroid of the triangle with vertices (1, 2, 3), (4, 5, 6) and (7, 8, 9).
  3. 6Find the point dividing A(−1, 2) and B(4, 7) in the ratio 2 : 3 internally.
Challenging
  1. 7M(2, 3, 4) is the midpoint of P(1, 1, 1) and Q. Find Q.
  2. 8Find the ratio in which the point (3, 4, 5) divides A(1, 2, 3) and B(7, 8, 9).
Show answers (try them yourself first)
  1. 1(3, 6, 8)
  2. 2(3, 4, 5)
  3. 3(2, 4, 6)
  4. 4(4, 4, 4)
  5. 5(4, 5, 6)
  6. 6(1, 4)
  7. 7Q = (3, 5, 7)
  8. 81 : 2 (internal)