# Vectors

## Overview

Welcome to the comprehensive course material on Vectors in Further Mathematics, a fundamental topic that serves as the building block for various concepts in mathematics and physics. In this course, we will delve deeply into understanding the essence of vectors and their applications, equipping you with the necessary skills to effectively manipulate and utilize vectors in problem-solving scenarios.

Concept of Vectors: To kickstart our journey, we will thoroughly explore the concept of vectors, elaborating on how they differ from scalar quantities and their significance in representing physical quantities that have both magnitude and direction. Understanding this foundational concept is crucial as it forms the basis for all vector operations.

Representation of Vectors: Moving forward, we will learn how to represent vectors in the form ai + bj, where 'a' and 'b' are the components of the vector along the x and y axes respectively. This form aids in visualizing vectors geometrically and performing arithmetic operations effectively.

Addition and Subtraction of Vectors: We will delve into the fundamental operations of vector addition and subtraction, exploring how vectors can be combined or separated to determine resultant vectors. Understanding the geometric interpretation of vector addition and subtraction is essential for solving complex problems involving multiple forces or velocities.

Multiplication of Vectors: In this course, we will not only cover the multiplication of vectors by scalars but also by other vectors. We will investigate how scalar multiplication affects the magnitude and direction of vectors and how vector multiplication yields new vectors perpendicular to the original vectors, opening doors to diverse applications in mathematics and physics.

Vector Laws: Triangle Law, Parallelogram Law, and Polygon Law are essential principles that govern vector operations. We will explore these laws to understand how vectors behave when arranged in various geometric configurations, enabling us to solve intricate problems involving forces, velocities, and displacements.

Diagrams and Problem-Solving: Visual aids and diagrams play a crucial role in understanding vector concepts. We will utilize diagrams to illustrate vector operations, enhancing our comprehension of vector properties and relationships. Additionally, we will tackle problems in elementary plane geometry, demonstrating how vectors can be applied to prove the concurrency of medians and diagonals in geometric figures.

This course aims to equip you with the necessary knowledge and skills to confidently work with vectors, unraveling their complexities, and harnessing their power to solve real-world problems. Get ready to embark on a fascinating journey through the realm of vectors in Further Mathematics!

## Objectives

1. Illustrate vector concepts through diagrams
2. Apply Triangle, Parallelogram and Polygon Laws in vector operations
3. Be able to represent vectors in the form ai + bj
4. Demonstrate the concurrency of medians and diagonals using vectors
5. Solve problems in elementary plane geometry using vectors
6. Solve equations involving vectors
7. Perform addition and subtraction of vectors
8. Understand the concept of vectors
9. Perform multiplication of vectors by vectors and scalars

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## Lesson Evaluation

Congratulations on completing the lesson on Vectors. Now that youve explored the key concepts and ideas, its time to put your knowledge to the test. This section offers a variety of practice questions designed to reinforce your understanding and help you gauge your grasp of the material.

You will encounter a mix of question types, including multiple-choice questions, short answer questions, and essay questions. Each question is thoughtfully crafted to assess different aspects of your knowledge and critical thinking skills.

Use this evaluation section as an opportunity to reinforce your understanding of the topic and to identify any areas where you may need additional study. Don't be discouraged by any challenges you encounter; instead, view them as opportunities for growth and improvement.

1. Represent the vector v = 3i + 4j in component form. A. (4, 3) B. (3, 4) C. (3) D. (4) Answer: A. (4, 3)
2. Find the resultant of vectors A = 2i + 3j and B = -i + 2j. A. i + 5j B. 3i + 5j C. i - j D. 3i + j Answer: D. 3i + j
3. If vector A = 5i - 2j and vector B = -3i + 7j, calculate 3A - 2B. A. 13i + 30j B. -1i + 18j C. 5i - 28j D. 11i + 20j Answer: A. 13i + 30j
4. Determine the magnitude of a vector v = 4i - 3j. A. 5 B. √7 C. 4 D. 3 Answer: A. 5
5. If vectors A = 3i + 4j and B = -2i - j, find the dot product of A and B. A. -10 B. -11 C. 11 D. 10 Answer: A. -10
6. Given vector A = 2i - 5j and B = 3i + 4j, calculate the cross product A x B. A. 23i + 10j B. 23i - 10j C. -23i + 10j D. -23i - 10j Answer: B. 23i - 10j
7. If vectors A = 5i - 3j and B = 2i + 7j, determine the angle between A and B. A. 35.97° B. 42.76° C. 58.13° D. 71.87° Answer: B. 42.76°
8. Find the unit vector in the direction of vector v = 4i - 2j. A. (2/√5)i - (1/√5)j B. (4/√20)i - (2/√20)j C. (4/√10)i - (2/√10)j D. (2/√10)i - (1/√10)j Answer: D. (2/√10)i - (1/√10)j
9. If vector A = 3i + j and vector B = -2i + 4j, determine the projection of A on B. A. 2i + j B. i - 2j C. -i + 2j D. -2i + j Answer: A. 2i + j

## Past Questions

Wondering what past questions for this topic looks like? Here are a number of questions about Vectors from previous years

Question 1

The vectors 6i + 8j and 8i - 6j are parallel to ?OP and ?OQ respectively. If the magnitude of ?OP and ?OQ are 80 units and 120 units respectively, express: ?OP and ?OQ in terms of i and j;

ii. |?PQ|, in the form c?k, where c and k are constants.

Practice a number of Vectors past questions