Prepared by Muhammad Tayyab, Subject Specialist Mathematics, Govt Christian High School Daska
📌 Based on National Curriculum 2023 / PECTAA 2026 Syllabus
📖 What's Inside: This exercise covers vector addition, scalar multiplication, magnitude, and component equations. Perfect for Punjab Boards exam preparation.
📚 Related Resources – Unit 6: Vectors in Plane
Class 10 Math Unit 6 Exercise 6.2 – Vector Operations: Complete Guide
Exercise 6.2 of Unit 6 takes the foundational concepts from Exercise 6.1 and moves them into vector operations. This exercise is where students learn to combine vectors through addition, scale them using scalar multiplication, find the magnitude of vector combinations, and solve for unknown components in vector equations. These skills are essential for the remainder of Unit 6 and form the basis of many problems in later chapters, especially Unit 7 (Trigonometry) where vectors are used with angles and directions.
What You Will Learn
By working through Exercise 6.2, students master the following operations: vector addition (adding corresponding components), scalar multiplication (multiplying a vector by a real number), combining both operations to simplify expressions, calculating the magnitude of combined vectors, and solving for unknown components in vector equations by equating coefficients of î and ĵ.
Topics Covered in This Exercise
- Vector addition – adding two or more vectors component-wise
- Scalar multiplication – multiplying a vector by a real scalar
- Combined operations – expressions like \(3\bar{a} + \frac{1}{2}\bar{b}\)
- Magnitude of vector combinations – finding \(|\bar{a} - 2\bar{b} + \bar{c}|\)
- Component equations – solving for \(x\) and \(y\) by equating î and ĵ coefficients
- Finding unknown vectors – using vector equations to solve for \(\bar{b}\)
Why Exercise 6.2 Is Important
Exercise 6.2 is the operational core of Unit 6. Board exam papers frequently feature questions that require students to add vectors, multiply by scalars, and compute magnitudes — and these operations appear in both short-answer and multi-part questions. Without the skills from Exercise 6.2, students cannot successfully tackle Exercise 6.3 or the Review Exercise. This exercise is also essential preparation for questions involving position vectors, which are tested in later units.
Punjab Board Preparation
Students preparing for board exams under any of the 10 BISE Punjab boards should prioritise this exercise. The combination of vector addition and scalar multiplication is one of the most common question types in the vectors chapter. The component-equation questions (like Q3 and Q6) are particularly popular with examiners because they test both understanding and algebra skills. Practising every part of Q1, Q2, and Q3 by hand is highly recommended.
PECTAA 2026 Syllabus Alignment
These solved notes are aligned with the Punjab Education & Curriculum Textbook Authority (PECTAA) 2026 syllabus, following the National Curriculum 2023 framework. The question numbering, formulas, and terminology match the current Punjab textbook exactly.
Exam Tips for Vector Operations
- Always group î and ĵ terms separately when simplifying.
- When finding magnitude of a combination, first simplify the vector fully, then apply the magnitude formula.
- For component equations, equate the coefficients of î and ĵ on both sides — this gives two equations that can be solved simultaneously.
- Double-check signs when subtracting vectors — a common source of errors.
Common Mistakes Students Make
- Forgetting to distribute a scalar to both components of a vector.
- Adding the coefficients of î and ĵ together incorrectly.
- Confusing \(|\bar{a} + \bar{b}|\) with \(|\bar{a}| + |\bar{b}|\) — these are not equal in general.
- Not simplifying final answers, especially fractions and surds.
Use the table of contents below to jump directly to any question, or scroll through the full solved exercise in order.
1 Vector Addition & Scalar Multiplication
Given \(\bar{a} = 7\hat{i} - 3\hat{j}\) and \(\bar{b} = \hat{i} + 5\hat{j}\)
(i) \(\bar{a} + \bar{b}\)
\[
\begin{aligned}
\bar{a} + \bar{b} &= (7\hat{i} - 3\hat{j}) + (\hat{i} + 5\hat{j}) \\
&= (7\hat{i} + \hat{i}) + (-3\hat{j} + 5\hat{j}) \\
&= 8\hat{i} + 2\hat{j}
\end{aligned}
\]
(ii) \(\bar{a} + 3\bar{b}\)
\[
\begin{aligned}
\bar{a} + 3\bar{b} &= (7\hat{i} - 3\hat{j}) + 3(\hat{i} + 5\hat{j}) \\
&= 7\hat{i} - 3\hat{j} + 3\hat{i} + 15\hat{j} \\
&= 10\hat{i} + 12\hat{j}
\end{aligned}
\]
(iii) \(3\bar{a} + \frac{1}{2}\bar{b}\)
\[
\begin{aligned}
3\bar{a} + \frac{1}{2}\bar{b} &= 3(7\hat{i} - 3\hat{j}) + \frac{1}{2}(\hat{i} + 5\hat{j}) \\
&= 21\hat{i} - 9\hat{j} + \frac{1}{2}\hat{i} + \frac{5}{2}\hat{j} \\
&= \frac{43}{2}\hat{i} - \frac{13}{2}\hat{j}
\end{aligned}
\]
(iv) \(\bar{b} - \bar{a}\)
\[
\begin{aligned}
\bar{b} - \bar{a} &= (\hat{i} + 5\hat{j}) - (7\hat{i} - 3\hat{j}) \\
&= \hat{i} + 5\hat{j} - 7\hat{i} + 3\hat{j} \\
&= -6\hat{i} + 8\hat{j}
\end{aligned}
\]
(v) \(4\bar{b} - 5\bar{a}\)
\[
\begin{aligned}
4\bar{b} - 5\bar{a} &= 4(\hat{i} + 5\hat{j}) - 5(7\hat{i} - 3\hat{j}) \\
&= 4\hat{i} + 20\hat{j} - 35\hat{i} + 15\hat{j} \\
&= -31\hat{i} + 35\hat{j}
\end{aligned}
\]
(vi) \(\frac{3}{2}\bar{a} - \bar{b}\)
\[
\begin{aligned}
\frac{3}{2}\bar{a} - \bar{b} &= \frac{3}{2}(7\hat{i} - 3\hat{j}) - (\hat{i} + 5\hat{j}) \\
&= \frac{21}{2}\hat{i} - \frac{9}{2}\hat{j} - \hat{i} - 5\hat{j} \\
&= \frac{19}{2}\hat{i} - \frac{19}{2}\hat{j}
\end{aligned}
\]
2 Magnitude of Vector Combinations
Given \(\bar{a} = 6\hat{i} - \hat{j}\), \(\bar{b} = \hat{i} + 5\hat{j}\), \(\bar{c} = 3\hat{i} + 5\hat{j}\)
(i) \(\bar{b} - \bar{c}\)
\[
\begin{aligned}
\bar{b} - \bar{c} &= (\hat{i} + 5\hat{j}) - (3\hat{i} + 5\hat{j}) \\
&= \hat{i} - 3\hat{i} + 5\hat{j} - 5\hat{j} \\
&= -2\hat{i} + 0\hat{j} \\
|\bar{b} - \bar{c}| &= \sqrt{(-2)^2 + 0^2} = 2
\end{aligned}
\]
(ii) \(\bar{a} - 2\bar{b} + \bar{c}\)
\[
\begin{aligned}
\bar{a} - 2\bar{b} + \bar{c} &= (6\hat{i} - \hat{j}) - 2(\hat{i} + 5\hat{j}) + (3\hat{i} + 5\hat{j}) \\
&= 6\hat{i} - \hat{j} - 2\hat{i} - 10\hat{j} + 3\hat{i} + 5\hat{j} \\
&= (6-2+3)\hat{i} + (-1-10+5)\hat{j} \\
&= 7\hat{i} - 6\hat{j} \\
|\bar{a} - 2\bar{b} + \bar{c}| &= \sqrt{7^2 + (-6)^2} = \sqrt{85}
\end{aligned}
\]
(iii) \(\bar{c} - \bar{b} - \bar{a}\)
\[
\begin{aligned}
\bar{c} - \bar{b} - \bar{a} &= (3\hat{i} + 5\hat{j}) - (\hat{i} + 5\hat{j}) - (6\hat{i} - \hat{j}) \\
&= 3\hat{i} + 5\hat{j} - \hat{i} - 5\hat{j} - 6\hat{i} + \hat{j} \\
&= (3-1-6)\hat{i} + (5-5+1)\hat{j} \\
&= -4\hat{i} + \hat{j} \\
|\bar{c} - \bar{b} - \bar{a}| &= \sqrt{(-4)^2 + 1^2} = \sqrt{17}
\end{aligned}
\]
3 Find x and y from Vector Equations
(i) \((x\hat{i} + y\hat{j}) + (2\hat{i} + 3\hat{j}) = 7\hat{i} + 6\hat{j}\)
\[
\begin{aligned}
(x\hat{i} + y\hat{j}) + (2\hat{i} + 3\hat{j}) &= 7\hat{i} + 6\hat{j} \\
x\hat{i} + y\hat{j} + 2\hat{i} + 3\hat{j} &= 7\hat{i} + 6\hat{j} \\
(x+2)\hat{i} + (y+3)\hat{j} &= 7\hat{i} + 6\hat{j}
\end{aligned}
\]
Equating components:
\[
\begin{aligned}
x+2 &= 7 \implies x = 5 \\
y+3 &= 6 \implies y = 3
\end{aligned}
\]
(ii) \((x\hat{i} - 5\hat{j}) + (3\hat{i} + 5\hat{j}) = -8\hat{i} + y\hat{j}\)
\[
\begin{aligned}
(x\hat{i} - 5\hat{j}) + (3\hat{i} + 5\hat{j}) &= -8\hat{i} + y\hat{j} \\
x\hat{i} + 3\hat{i} - 5\hat{j} + 5\hat{j} &= -8\hat{i} + y\hat{j} \\
(x+3)\hat{i} + 0\hat{j} &= -8\hat{i} + y\hat{j}
\end{aligned}
\]
Equating components:
\[
\begin{aligned}
x+3 &= -8 \implies x = -11 \\
0 &= y \implies y = 0
\end{aligned}
\]
(iii) \((y\hat{i} + 3\hat{j}) + (-5\hat{i} + 2x\hat{j}) = 9\hat{i} + 7\hat{j}\)
\[
\begin{aligned}
(y\hat{i} + 3\hat{j}) + (-5\hat{i} + 2x\hat{j}) &= 9\hat{i} + 7\hat{j} \\
y\hat{i} - 5\hat{i} + 3\hat{j} + 2x\hat{j} &= 9\hat{i} + 7\hat{j} \\
(y-5)\hat{i} + (3+2x)\hat{j} &= 9\hat{i} + 7\hat{j}
\end{aligned}
\]
Equating components:
\[
\begin{aligned}
y-5 &= 9 \implies y = 14 \\
3+2x &= 7 \implies 2x = 4 \implies x = 2
\end{aligned}
\]
4 Find Magnitude of \(\bar{b}\)
Given \(\bar{a} = \hat{i} + 3\hat{j}\), \(\bar{c} = 2\hat{i} + \hat{j}\) and \(\bar{a} + 2\bar{b} = \bar{c}\)
\[
\begin{aligned}
\bar{a} + 2\bar{b} &= \bar{c} \\
(\hat{i} + 3\hat{j}) + 2\bar{b} &= 2\hat{i} + \hat{j} \\
2\bar{b} &= (2\hat{i} + \hat{j}) - (\hat{i} + 3\hat{j}) \\
2\bar{b} &= \hat{i} - 2\hat{j} \\
\bar{b} &= \frac{1}{2}\hat{i} - \hat{j}
\end{aligned}
\]
Now, magnitude:
\[
|\bar{b}| = \sqrt{\left(\frac{1}{2}\right)^2 + (-1)^2} = \sqrt{\frac{1}{4} + 1} = \sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}
\]
5 Vector Components of \(\bar{a} + 5\bar{b}\)
Given \(\bar{a} = -2\hat{i} + 7\hat{j}\), \(\bar{b} = 3\hat{i} - 5\hat{j}\)
\[
\begin{aligned}
\bar{a} + 5\bar{b} &= (-2\hat{i} + 7\hat{j}) + 5(3\hat{i} - 5\hat{j}) \\
&= -2\hat{i} + 7\hat{j} + 15\hat{i} - 25\hat{j} \\
&= 13\hat{i} - 18\hat{j}
\end{aligned}
\]
6 Find Values of m and n
Given \(5\hat{i} - 3\hat{j} = m(\hat{i} - 10\hat{j}) + n(4\hat{i} - 3\hat{j})\)
\[
\begin{aligned}
5\hat{i} - 3\hat{j} &= m(\hat{i} - 10\hat{j}) + n(4\hat{i} - 3\hat{j}) \\
&= m\hat{i} - 10m\hat{j} + 4n\hat{i} - 3n\hat{j} \\
&= (m+4n)\hat{i} + (-10m-3n)\hat{j}
\end{aligned}
\]
Equating components:
\[
\begin{aligned}
m + 4n &= 5 \quad \text{(i)} \\
-10m - 3n &= -3 \quad \text{(ii)}
\end{aligned}
\]
From (i): \(m = 5 - 4n\). Substituting in (ii):
\[
\begin{aligned}
-10(5 - 4n) - 3n &= -3 \\
-50 + 40n - 3n &= -3 \\
37n &= 47 \\
n &= \frac{47}{37}
\end{aligned}
\]
Putting \(n\) in equation (i):
\[
\begin{aligned}
m + 4\left(\frac{47}{37}\right) &= 5 \\
m &= 5 - \frac{188}{37} = \frac{185 - 188}{37} = -\frac{3}{37}
\end{aligned}
\]
📈 Key Concepts & Quick Revision
- Vector Addition: Add corresponding components: \((x_1\hat{i} + y_1\hat{j}) + (x_2\hat{i} + y_2\hat{j}) = (x_1+x_2)\hat{i} + (y_1+y_2)\hat{j}\)
- Scalar Multiplication: \(k(x\hat{i} + y\hat{j}) = kx\hat{i} + ky\hat{j}\)
- Magnitude: \(|\bar{a}| = \sqrt{x^2 + y^2}\)
- Vector Equality: Two vectors are equal if their corresponding components are equal.
🎯 Important Definitions
- Vector: a quantity with both magnitude and direction.
- Scalar: a real number used to scale a vector.
- Resultant Vector: the sum of two or more vectors.
📝 Important MCQs for Practice
1. If \(\bar{a} = 2\hat{i} + 3\hat{j}\) and \(\bar{b} = 4\hat{i} - \hat{j}\), then \(\bar{a} + \bar{b}\) equals:
2. If \(\bar{a} = 3\hat{i} - 4\hat{j}\), then \(|\bar{a}|\) equals:
3. If \(\bar{a} = 2\hat{i} - 3\hat{j}\), then \(2\bar{a}\) equals:
🏆 Board Exam Tips & Strategy
- Show all steps when simplifying vector expressions — examiners award marks for intermediate steps.
- For component equations, write out the equation clearly before equating coefficients.
- Leave answers in exact surd form unless a decimal is required.
- Practice simultaneous equations for component problems — they appear frequently.
❓ Frequently Asked Questions
What is taught in Exercise 6.2 of Class 10 Math Unit 6?
Exercise 6.2 covers vector addition, scalar multiplication, magnitude of vector combinations, and solving component equations.
How many questions are there in Unit 6 Exercise 6.2?
Exercise 6.2 has 6 main questions covering vector operations and component equations.
Is this solution according to the PECTAA 2026 syllabus?
Yes, these solutions are prepared according to the PECTAA 2026 / National Curriculum 2023 syllabus.
Is this Exercise 6.2 solution valid for all Punjab Boards?
Yes, the content follows the unified Punjab textbook and is applicable to all 10 BISE Punjab boards.
What is vector addition?
Vector addition is the operation of adding two vectors component-wise: \((x_1î + y_1ĵ) + (x_2î + y_2ĵ) = (x_1+x_2)î + (y_1+y_2)ĵ\).
What is scalar multiplication of a vector?
Scalar multiplication is multiplying a vector by a real number: \(k(xî + yĵ) = kxî + kyĵ\).
How do you find the magnitude of a vector combination?
First combine the vectors, then use the formula \(|\bar{a}| = \sqrt{x^2 + y^2}\).
Are solved PDF notes available for Exercise 6.2?
Yes, a complete solved PDF is embedded on this page and available for free download.
Can I download the Unit 6 Exercise 6.2 solution as a PDF?
Yes, use the Download PDF button on this page to save the complete solved Exercise 6.2 notes.
Is Exercise 6.2 important for Class 10 board exams?
Yes, vector operations are frequently tested, and this exercise builds essential skills for later units.
Who prepared these Class 10 Math Unit 6 notes?
These notes were prepared by Muhammad Tayyab, Subject Specialist Mathematics at Govt Christian High School Daska.