Prepared by Muhammad Tayyab, Subject Specialist Mathematics, Govt Christian High School Daska
📌 Based on National Curriculum 2023 / PECTAA 2026 Syllabus
📖 What's Inside: This review exercise covers MCQs, magnitude of vectors, unit vector, vector operations, translations, parallelogram verification, isosceles triangle using vector magnitude, projectile motion, resultant velocity, and conceptual questions. Perfect for Punjab Boards exam preparation.
📚 Related Resources – Unit 6: Vectors in Plane
Mastering Vectors: Class 10 Math Unit 6 Review Exercise
The Review Exercise of Unit 6 is your ultimate revision tool for Vectors in Plane, covering all the key concepts from Exercises 6.1 through 6.3. This comprehensive review is designed to solidify your understanding of vector operations, translations, and geometric applications, ensuring you are fully prepared for the Punjab Board exams. By working through this exercise, you will consolidate your ability to calculate magnitudes, find unit vectors, perform vector operations, and apply vectors to solve problems involving parallelograms, isosceles triangles, and real-world scenarios like projectile motion.
What You Will Learn in This Review
This review exercise is structured to test and reinforce your knowledge across the entire unit. You will revisit the fundamental concept of a vector as a quantity with both magnitude and direction, and learn to represent them in component form. You will practice finding the magnitude and direction of a vector, determining unit vectors, and performing addition, subtraction, and scalar multiplication. The exercise also covers important geometric applications: using vectors to translate figures, verifying parallelograms, and proving triangles are isosceles. Furthermore, you will apply vectors to solve problems involving velocity, including resultant velocity and projectile motion, bridging the gap between abstract mathematics and physics.
Topics Covered in This Review
- MCQs – Testing fundamental concepts of vectors and their properties.
- Magnitude of Vectors – Calculating the length of a vector using the distance formula.
- Unit Vectors – Finding a vector of magnitude 1 in the same direction as a given vector.
- Vector Operations – Addition, subtraction, and scalar multiplication of vectors.
- Translations – Using vectors to translate points and figures in the plane.
- Parallelogram Verification – Proving a quadrilateral is a parallelogram using vector equality.
- Isosceles Triangle – Using vector magnitudes to prove a triangle is isosceles.
- Resultant Velocity – Finding the resultant vector of two velocities and its magnitude.
- Conceptual Questions – Deepening understanding of equality and opposites of vectors.
Why This Review Exercise is Crucial for Board Exams
The Review Exercise is not just another set of problems; it is a carefully curated collection that mirrors the style and difficulty of questions that appear in board exams. It provides a holistic view of the unit, helping you identify your strengths and weaknesses. By mastering this review, you'll gain the confidence to tackle any vector-related question, from straightforward calculations to complex geometric and application-based problems, ensuring you secure top marks in your mathematics paper.
📖 Multiple Choice Questions (Unit 6 Review)
1. x-axis and y-axis divide a coordinate plane into ___ parts.
✅ Correct Answer: (D) four
x-axis and y-axis divide the coordinate plane into four quadrants.
x-axis and y-axis divide the coordinate plane into four quadrants.
2. P(4, -4) lies in ___ quadrant.
✅ Correct Answer: (D) fourth
P(4, -4) has x > 0 and y < 0, so it lies in the fourth quadrant.
P(4, -4) has x > 0 and y < 0, so it lies in the fourth quadrant.
3. A vector having magnitude 1, is called:
✅ Correct Answer: (C) unit vector
A vector with magnitude 1 is called a unit vector.
A vector with magnitude 1 is called a unit vector.
4. What is the value of \(|3i + 4j|\)?
✅ Correct Answer: (C) 5
\(|3i + 4j| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\)
\(|3i + 4j| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\)
5. If \(a = \lambda b\), then \(a\) and \(b\) are:
✅ Correct Answer: (B) parallel
If \(a = \lambda b\), then \(a\) and \(b\) are parallel vectors.
If \(a = \lambda b\), then \(a\) and \(b\) are parallel vectors.
6. If \(\overline{OA} = a\), \(\overline{OB} = b\), then \(\overline{AB}\) is:
✅ Correct Answer: (A) \(b - a\)
\(\overline{AB} = \overline{OB} - \overline{OA} = b - a\)
\(\overline{AB} = \overline{OB} - \overline{OA} = b - a\)
7. Translation vector shows:
✅ Correct Answer: (C) movement
Translation vector shows movement (displacement).
Translation vector shows movement (displacement).
8. Sum of two vectors is:
✅ Correct Answer: (B) a vector
Sum of two vectors is always a vector.
Sum of two vectors is always a vector.
9. The position vector of point P(3, -2) with respect to O is:
✅ Correct Answer: (B) \(3i - 2j\)
Position vector of P(3, -2) is \(3i - 2j\) (from origin to point).
Position vector of P(3, -2) is \(3i - 2j\) (from origin to point).
10. Vector from point P(3,4) to origin is:
✅ Correct Answer: (C) \(-3i - 4j\)
Vector from P(3,4) to origin = O - P = \((0i+0j) - (3i+4j) = -3i - 4j\)
Vector from P(3,4) to origin = O - P = \((0i+0j) - (3i+4j) = -3i - 4j\)
📋 Answer Key with Explanations
| Q. No | Correct Answer | Explanation |
|---|---|---|
| 1 | D | x-axis and y-axis divide the coordinate plane into four quadrants. |
| 2 | D | P(4, -4) has x > 0 and y < 0, so it lies in the fourth quadrant. |
| 3 | C | A vector with magnitude 1 is called a unit vector. |
| 4 | C | \(|3i + 4j| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\) |
| 5 | B | If \(a = \lambda b\), then \(a\) and \(b\) are parallel vectors. |
| 6 | A | \(\overline{AB} = \overline{OB} - \overline{OA} = b - a\) |
| 7 | C | Translation vector shows movement (displacement). |
| 8 | B | Sum of two vectors is always a vector. |
| 9 | B | Position vector of P(3, -2) is \(3i - 2j\) (from origin to point). |
| 10 | C | Vector from P(3,4) to origin = O - P = \(-3i - 4j\) |
Prepared By: M. Tayyab, SSE (Math) Govt Christian High School, Daska.
2 Find magnitude of the AB
(i) A(7,7), B(-12,0)
Vector AB:
\[
\begin{aligned}
\overline{AB} &= \overline{OB} - \overline{OA} \\
&= (-12i + 0j) - (7i + 7j) \\
&= -12i - 7i - 7j \\
&= -19i - 7j
\end{aligned}
\]
Magnitude:
\[
|\overline{AB}| = \sqrt{(-19)^2 + (-7)^2} = \sqrt{361 + 49} = \sqrt{410}
\]
(ii) A(9,3), B(2,11)
\[
\begin{aligned}
\overline{AB} &= (2i + 11j) - (9i + 3j) = -7i + 8j \\
|\overline{AB}| &= \sqrt{(-7)^2 + 8^2} = \sqrt{49 + 64} = \sqrt{113}
\end{aligned}
\]
Hence, \( |\overline{AB}| = \sqrt{410} \) and \( \sqrt{113} \) respectively.
3 Find a unit vector in the direction of \(\overline{a} = \frac{5}{3} i + \frac{1}{3} j\)
\[
|\overline{a}| = \sqrt{\left(\frac{5}{3}\right)^2 + \left(\frac{1}{3}\right)^2} = \sqrt{\frac{25}{9} + \frac{1}{9}} = \sqrt{\frac{26}{9}} = \frac{\sqrt{26}}{3}
\]
Unit vector:
\[
\hat{a} = \frac{\overline{a}}{|\overline{a}|} = \frac{\frac{5}{3}i + \frac{1}{3}j}{\frac{\sqrt{26}}{3}} = \frac{5}{\sqrt{26}}i + \frac{1}{\sqrt{26}}j
\]
Hence, \( \hat{a} = \frac{5}{\sqrt{26}}i + \frac{1}{\sqrt{26}}j \).
4 Vector Operations
If \(\overline{a} = 2i - j\), \(\overline{b} = 3i + j\) and \(\overline{c} = 4i + j\), then find:
(i) \(5\overline{b} - \overline{a} + \overline{c}\)
\[
\begin{aligned}
\overline{v} &= 5(3i + j) - (2i - j) + (4i + j) \\
&= 15i + 5j - 2i + j + 4i + j \\
&= 17i + 7j
\end{aligned}
\]
(ii) \(8\overline{a} + \overline{b} + 5\overline{c}\)
\[
\begin{aligned}
\overline{v} &= 8(2i - j) + (3i + j) + 5(4i + j) \\
&= 16i - 8j + 3i + j + 20i + 5j \\
&= 39i - 2j
\end{aligned}
\]
(iii) \(\overline{c} + \overline{b} - 4\overline{a}\)
\[
\begin{aligned}
\overline{v} &= (4i + j) + (3i + j) - 4(2i - j) \\
&= 4i + j + 3i + j - 8i + 4j \\
&= -i + 6j
\end{aligned}
\]
Hence, \(17i+7j\), \(39i-2j\), and \(-i+6j\) respectively.
5 Find the values of \(x\) and \(y\)
\[(2x i + y j) + (-i + 5j) = \frac{1}{4} i - 8j\]
\[
\begin{aligned}
(2x i + y j) + (-i + 5j) &= \frac{1}{4} i - 8j \\
(2x - 1)i + (y + 5)j &= \frac{1}{4} i - 8j
\end{aligned}
\]
Equating components:
\[
2x - 1 = \frac{1}{4} \implies 2x = \frac{5}{4} \implies x = \frac{5}{8}
\]
\[
y + 5 = -8 \implies y = -13
\]
Hence, \(x = \frac{5}{8}\) and \(y = -13\).
6 Translate Triangle
Plot \(A(-5,3)\), \(B(-2,3)\) and \(C(-4,5)\) to form triangle ABC. Translate by vector \(5i - 2j\).
\[
\begin{aligned}
A' &= (-5i + 3j) + (5i - 2j) = 0i + j = (0,1) \\
B' &= (-2i + 3j) + (5i - 2j) = 3i + j = (3,1) \\
C' &= (-4i + 5j) + (5i - 2j) = i + 3j = (1,3)
\end{aligned}
\]
Hence, translated triangle vertices are \(A'(0,1)\), \(B'(3,1)\), and \(C'(1,3)\).
7 Verify Parallelogram ABCD
Use vectors to show that ABCD is a parallelogram, where \(A(2,3)\), \(B(6,3)\), \(C(7,6)\), \(D(3,6)\).
\[
\begin{aligned}
\overline{AB} &= (6i+3j) - (2i+3j) = 4i + 0j \\
\overline{DC} &= (7i+6j) - (3i+6j) = 4i + 0j \\
\overline{AD} &= (3i+6j) - (2i+3j) = i + 3j \\
\overline{BC} &= (7i+6j) - (6i+3j) = i + 3j
\end{aligned}
\]
Since \(\overline{AB} = \overline{DC}\) and \(\overline{AD} = \overline{BC}\), ABCD is a parallelogram.
8 Isosceles Triangle ABC
Use vectors to show that triangle ABC is isosceles, where \(A(1,2)\), \(B(4,6)\), \(C(7,2)\).
\[
\begin{aligned}
\overline{AB} &= 3i + 4j, & |\overline{AB}| &= 5 \\
\overline{BC} &= 3i - 4j, & |\overline{BC}| &= 5 \\
\overline{AC} &= 6i + 0j, & |\overline{AC}| &= 6
\end{aligned}
\]
Since \(|\overline{AB}| = |\overline{BC}| = 5\), triangle ABC is isosceles.
9 Magnitude of Velocity
A ball is projected with velocity vector \(\overline{v} = 6i + 8j\). What is the magnitude of velocity?
\[
|\overline{v}| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
Hence, magnitude of velocity = 10 units.
10 Resultant Velocity and Magnitude
An aircraft is flying due east with airspeed \(200km/h\). There is a wind blowing due north at \(60km/h\). Find the resultant velocity and its magnitude.
\[
\begin{aligned}
\overline{v}_p &= 200i \, km/h \\
\overline{v}_w &= 60j \, km/h \\
\overline{v}_r &= \overline{v}_p + \overline{v}_w = 200i + 60j \\
|\overline{v}_r| &= \sqrt{200^2 + 60^2} = \sqrt{40000 + 3600} = \sqrt{43600} \approx 208.81 \, km/h
\end{aligned}
\]
Resultant velocity = \(200i + 60j\), Magnitude ≈ 208.81 km/h.
11 Conceptual Questions
(i) Suppose vectors \(\overline{a}\) and \(\overline{b}\) are equal. Can we say they originate from the same point? Why or why not?
No. Equal vectors may have different initial points, provided they have the same magnitude and direction.
(ii) Do they have equal magnitudes? Explain.
Yes. Equal vectors always have equal magnitudes.
(iii) Do they have same direction? Why?
Yes. Equal vectors have the same direction as well as the same magnitude.
(i) Suppose vectors \(\overline{a}\) and \(\overline{b}\) are opposite. Can we assume they begin at the same point? Give a reason.
No. Opposite vectors may have different initial points.
(ii) Do they have same magnitude? Why?
Yes. Opposite vectors have equal magnitudes.
(iii) Do they have the same direction? Explain why or why not.
No. Opposite vectors have opposite directions.
📈 Key Concepts & Quick Revision
- Vector Translation: Add translation vector to position vector of each vertex.
- Parallelogram Property: Opposite sides are equal and parallel: \(\overline{AB} = \overline{DC}\).
- Isosceles Triangle: At least two side vectors have equal magnitude.
- Unit Vector: \(\hat{a} = \frac{\overline{a}}{|\overline{a}|}\).
- Resultant Velocity: Add vectors component-wise.
- Equal Vectors: Same magnitude and direction; may have different initial points.
- Opposite Vectors: Same magnitude, opposite direction.
❓ Frequently Asked Questions
What is covered in Unit 6 Review Exercise?
The review exercise covers MCQs, magnitude of vectors, unit vector, vector operations, translations, parallelogram verification, isosceles triangle using vector magnitude, projectile motion, resultant velocity, and conceptual questions.
How many MCQs are in Unit 6 Review Exercise?
There are 10 multiple choice questions covering key concepts from Unit 6.
Is this solution according to the PECTAA 2026 syllabus?
Yes, these solutions are prepared according to the PECTAA 2026 / National Curriculum 2023 syllabus.
Are solved PDF notes available for Review Exercise 6?
Yes, a complete solved PDF is embedded on this page and available for free download.
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.