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medium difficulty9 min

Section Formula in Vector Form

The section formula gives the position vector of a point R that divides the line segment joining points P and Q in a given ratio. If position vectors of P and Q are p and q, and R divides PQ internally in ratio m:n, then r = (mp? no, wait)

In one line: The nearer endpoint gets the smaller opposite weight; for AP:PB = m:n, use n with a and m with b.

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Understand it

For internal division, the point lies between P and Q. For external division, the point lies outside the segment. The formula must be applied with the correct order of ratio and endpoints.

Example

If P has position vector p and Q has position vector q, and R divides PQ internally in ratio 2:3, then r = (2q + 3p)/5.

Formula and rule

Internal section formula: if R divides PQ in ratio m:n, then r = (np + mq)/(m + n).

External section formula: r = (mq - np)/(m - n), with careful sign use and m not equal to n for this form.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Let A have position vector a = 2i - j + 3k and B have b = 5i + 2j - k.

  1. 1Find the position vector of P dividing AB internally in ratio 2:1.
  2. 2Here AP:PB = 2:1, so p = (1a + 2b)/(2 + 1).
  3. 3Substitute: p = [(2i - j + 3k) + 2(5i + 2j - k)]/3 = (2i - j + 3k + 10i + 4j - 2k)/3 = (12i + 3j + k)/3 = 4i + j + (1/3)k.

Method you can reuse

Identify the endpoints and their position vectors; write the ratio in the order first segment:second segment; choose internal or external formula; substitute with correct weights; simplify each component separately.

Visual tip

A simple line-segment diagram can help track P, R, Q and the ratio PR:RQ, but component substitution is the main exam method.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Point R divides AB internally in ratio 1:2. If a = i + j and b = 4i + 7j, find r.

Think first, then show answer
r = (2a + 1b)/3 = (2(i + j) + 4i + 7j)/3 = (6i + 9j)/3 = 2i + 3j.

Common wrong answer

Using r = (ma + nb)/(m + n) for AP:PB = m:n, which reverses the weights and gives an incorrect point.

Don't mix this up

Students often reverse p and q in the formula, which gives the point on the wrong side of the segment.

Exam tip

If the ratio is PR:RQ = m:n, then the weight near P is n and the weight near Q is m for internal division: r = (np + mq)/(m + n).

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Usually asked as a direct position-vector calculation, midpoint problem, ratio-finding problem, or coordinate equivalent.
  1. 1

    Enough for 1 mark

    The section formula gives the position vector of a point R that divides the line segment joining points P and Q in a given ratio. If position vectors of P and Q are p and q, and R divides PQ internally in ratio m:n, then r = (mp? no, wait).

  2. 2

    Add for 2 marks

    Example: If P has position vector p and Q has position vector q, and R divides PQ internally in ratio 2:3, then r = (2q + 3p)/5.

  3. 3

    Add for 3 marks

    For internal division, the point lies between P and Q.

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