Class 10 Maths Chapter 6 Exercise 6.3 Solutions 2026 – Vectors in Plane Notes PDF

Unit 6: Vectors in Plane | Exercise 6.3 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 6.3 solutions for Vectors in Plane have been uploaded below and are free to view or download, fully solved according to the new PCTB/PECTAA Class 10 Mathematics syllabus.

Using Vectors to Describe Geometry

With magnitude, direction, and the basic operations already established, Exercise 6.3 turns to how vectors describe geometric relationships — finding a unit vector in a given direction, and locating a point that divides a line segment in a given ratio. These ideas connect vector algebra directly back to coordinate geometry.

Key Concepts Covered in Exercise 6.3

Unit Vectors in a Given Direction

A unit vector has magnitude exactly 1. To find the unit vector in the same direction as a given vector a, divide a by its own magnitude: â = a / |a|. This preserves the direction while scaling the length to exactly 1.

The Standard Unit Vectors i and j

Any vector (x, y) in the plane can be written as xi + yj, where i = (1, 0) and j = (0, 1) are the standard unit vectors along the x-axis and y-axis. This notation makes vector components explicit.

The Section Formula

If a point P divides the line segment joining points A and B in the ratio m:n, its position vector is p = (n·a + m·b) / (m+n), where a and b are the position vectors of A and B. This includes the midpoint as the special case where m = n = 1.

The Midpoint of a Line Segment

As a special case of the section formula, the midpoint M of segment AB has position vector m = (a + b) / 2 — simply the average of the two position vectors.

Step-by-Step Solved Examples

Q. No. 1: Find the unit vector in the direction of a = (3, 4).

|a| = √(3²+4²) = √25 = 5

â = a/|a| = (3/5, 4/5)

Q. No. 2: Find the midpoint of the segment joining A(2, 3) and B(6, 7).

Midpoint = (a+b)/2 = [(2,3)+(6,7)] / 2 = (8,10)/2

Midpoint = (4, 5)

Q. No. 3: Find the position vector of the point P that divides the segment joining A(1, 2) and B(4, 8) in the ratio 1:2.

p = (n·a + m·b)/(m+n), with m=1, n=2

p = [2(1,2) + 1(4,8)] / (1+2)

p = [(2,4) + (4,8)] / 3 = (6,12)/3

p = (2, 4)

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 6.3 in the same numbered, step-by-step format.

MCQs, Short Questions & Long Questions from This Exercise

Alongside the main exercise, this page’s uploaded notes are organized the way Punjab Board papers are structured, so students can practice by question type:

• MCQs: Quick questions on writing a given vector in terms of i and j
• Short Questions: Finding the unit vector of a given vector, like Q. No. 1 above
• Long Questions: Applying the section formula to find a dividing point, like Q. No. 3 above

Where This Matters Beyond This Exercise

The section formula introduced here is the vector equivalent of concepts covered in coordinate geometry, and reinforces how the same geometric idea can be approached from different branches of mathematics. Unit vectors are also essential in physics for describing pure direction, separate from magnitude, such as in stating the direction of a force or velocity.

Common Mistakes Students Make in Exercise 6.3

• Dividing only one component by the magnitude when finding a unit vector, instead of both
• Mixing up m and n in the section formula, which reverses the ratio
• Forgetting that the midpoint formula is just the section formula with a 1:1 ratio
• Making an arithmetic slip when adding position vectors before dividing

Why This Exercise Matters for the Board Exam

The section formula is a compact, frequently tested long-question type, since it combines vector addition, scalar multiplication, and careful substitution of the ratio in one question. Unit vector questions are shorter but equally common, making both topics worth practicing until the formulas are second nature.

Quick Links – Chapter 6: Vectors in Plane

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 6.3 notes free to download?

Yes, all notes on this page are completely free to view and download in PDF format.

Q2. Which board are these notes for?

These notes are prepared according to the Punjab Board syllabus and are useful for all Punjab boards (Lahore, Gujranwala, Multan, Sargodha, Rawalpindi, Faisalabad, DG Khan, Bahawalpur, Sahiwal) as well as the Federal Board (FBISE).

Q3. What is the difference between the section formula and the midpoint formula?

The midpoint formula is simply a special case of the section formula, used when a point divides a segment exactly in half (a 1:1 ratio). The section formula works for any ratio m:n, not just equal halves.

Q4. Does this page include MCQs and short questions too?

Yes, the uploaded notes include MCQs, short questions, and long questions related to this exercise’s topics, organized by question type to match the board paper pattern.

Q5. How can I download the PDF?

Click the “Download PDF” button above and the notes will open or download directly to your device.

Q6. Are these notes updated for the new 2026–27 syllabus?

Yes, these notes are prepared strictly according to the latest PCTB/PECTAA syllabus for the 2026–27 academic session.

Comments & Feedback

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