UK K12 GCSE/A-Level: Year 11 GCSE Mathematics - Vectors, Notation and Geometry Proofs

Learning objectives

By the end of this topic, students will be able to:

• Understand the notation and geometry of vectors, including the concept of a vector as a directed line segment.
• Apply the concept of vector addition and scalar multiplication to solve problems.
• Prove geometrically that the sum of vectors is commutative and associative.
• Use the properties of vectors to solve problems in a variety of contexts.

Core concepts

Vectors are mathematical objects that have both magnitude (length) and direction. They can be represented graphically as arrows or directed line segments. The notation for vectors typically involves boldface type, with the vector's components written in a row or column.

Vector addition

Vector addition is the process of combining two or more vectors to obtain a resultant vector. The resultant vector is the sum of the individual vectors. Geometrically, vector addition can be represented by drawing the vectors head-to-tail, with the resultant vector extending from the tail of the first vector to the head of the last vector.

Scalar multiplication

Scalar multiplication is the process of multiplying a vector by a scalar (a number). The resulting vector has a magnitude that is the product of the original vector's magnitude and the scalar, and a direction that is the same as the original vector.

Commutativity and associativity of vector addition

The commutative property of vector addition states that the order of the vectors being added does not affect the resultant vector. In other words, A + B = B + A, where A and B are vectors.

The associative property of vector addition states that the order in which vectors are added does not affect the resultant vector. In other words, (A + B) + C = A + (B + C), where A, B, and C are vectors.

Proof of commutativity

To prove that vector addition is commutative, we can use the following diagram:

  A
 / \
/   \
B  |  C
 \ /
  A + B + C

Since the order of the vectors being added does not affect the resultant vector, we can conclude that A + B = B + A.

Proof of associativity

To prove that vector addition is associative, we can use the following diagram:

  A
 / \
/   \
B  |  C
 \ /
  A + B + C

Since the order in which vectors are added does not affect the resultant vector, we can conclude that (A + B) + C = A + (B + C).

Worked examples

Example 1: Vector addition

Find the resultant vector of A + B, where A = 2i + 3j and B = 4i - 2j.

Solution

To find the resultant vector, we add the corresponding components of the two vectors:

A + B = (2 + 4)i + (3 - 2)j = 6i + j

Example 2: Scalar multiplication

Find the vector resulting from multiplying A by 2, where A = 3i - 2j.

Solution

To find the resulting vector, we multiply the corresponding components of the vector by the scalar:

2A = 2(3i - 2j) = 6i - 4j

Example 3: Commutativity of vector addition

Show that A + B = B + A, where A = 2i + 3j and B = 4i - 2j.

Solution

To show that A + B = B + A, we can use the following diagram:

  A
 / \
/   \
B  |  C
 \ /
  A + B + C

Since the order of the vectors being added does not affect the resultant vector, we can conclude that A + B = B + A.

Common misconceptions

• Vectors are only defined in two dimensions.
• Vector addition is not commutative.
• Scalar multiplication is not associative.

Exam tips

• Make sure to use the correct notation for vectors, including boldface type and components written in a row or column.
• Be careful when applying the commutative and associative properties of vector addition.
• Use diagrams to help visualize vector addition and scalar multiplication.

MCQs with explanations

MCQ 1: [F]

What is the resultant vector of A + B, where A = 2i + 3j and B = 4i - 2j?

A) 6i + j B) 6i - j C) 10i + j D) 10i - j

Correct answer: A) 6i + j Why the distractors fail: B) 6i - j is the resultant vector of A - B, not A + B. C) 10i + j is the resultant vector of A + 2B, not A + B. D) 10i - j is the resultant vector of A - 2B, not A + B.

MCQ 2: [H]

Show that (A + B) + C = A + (B + C), where A = 2i + 3j, B = 4i - 2j, and C = 6i + 9j.

A) True B) False

Correct answer: A) True Why the distractors fail: B) False is not a valid answer choice.

MCQ 3: [F]

What is the vector resulting from multiplying A by 2, where A = 3i - 2j?

A) 6i - 4j B) 6i + 4j C) 3i - 4j D) 3i + 4j

Correct answer: A) 6i - 4j Why the distractors fail: B) 6i + 4j is the vector resulting from multiplying A by -2, not 2. C) 3i - 4j is the original vector A, not the resulting vector. D) 3i + 4j is not a valid vector.

MCQ 4: [H]

Show that A + B = B + A, where A = 2i + 3j and B = 4i - 2j.

A) True B) False

Correct answer: A) True Why the distractors fail: B) False is not a valid answer choice.

MCQ 5: [F]

What is the resultant vector of A + B, where A = 2i + 3j and B = 4i + 2j?

A) 6i + 5j B) 6i - 5j C) 10i + 5j D) 10i - 5j

Correct answer: A) 6i + 5j Why the distractors fail: B) 6i - 5j is the resultant vector of A - B, not A + B. C) 10i + 5j is the resultant vector of A + 2B, not A + B. D) 10i - 5j is the resultant vector of A - 2B, not A + B.

Short-answer questions

Question 1

Show that (A + B) + C = A + (B + C), where A = 2i + 3j, B = 4i - 2j, and C = 6i + 9j.

Question 2

Find the resultant vector of A + B, where A = 2i + 3j and B = 4i - 2j.

Question 3

Show that A + B = B + A, where A = 2i + 3j and B = 4i - 2j.

Question 4

Find the vector resulting from multiplying A by 2, where A = 3i - 2j.

Question 5

Show that (A + B) + C = A + (B + C), where A = 2i + 3j, B = 4i - 2j, and C = 6i + 9j.