Explain how a vector can be visualised as an arrow.

Draw the arrow so that it begins at the origin and finishes at the point whose coordinates are the vector's entries. The 2-vector [2.0, 3.0] over ℝ therefore becomes the arrow running from (0, 0) to (2.0, 3.0). Once drawn this way, the arrow's length carries the vector's magnitude and the way it points carries its direction. A 3-vector is pictured identically, with the arrow reaching from the origin out to a point in three-dimensional space.

How are two vectors added?

Entry by entry: the entry at each position of the result is the sum of the entries at that position in the two vectors. The vectors must have the same number of entries. For example [2, 3] + [4, 1] = [2 + 4, 3 + 1] = [6, 4].

What does vector addition achieve geometrically?

Translation. Adding a vector to a point moves that point by the amount the added vector describes, without changing its orientation or size. Adding [4, 1] to the point [2, 3] moves it 4 to the right and 1 up, giving [6, 4]. Drawn as arrows, the sum is found by placing the tail of the second arrow at the head of the first.

How is a vector multiplied by a scalar?

Every entry of the vector is multiplied by the scalar. For example 2 x [3, -1] = [6, -2], and 0.5 x [4, 8] = [2, 4].

What does scalar-vector multiplication achieve geometrically?

Scaling. Multiplying by a scalar changes the length of the arrow by that factor while leaving its direction along the same line: a scalar greater than 1 stretches it, a scalar between 0 and 1 shrinks it, and a negative scalar reverses the direction as well as scaling the length by the size of the scalar.

Calculate 3 x [2, -1, 4] and then add [1, 1, 1].

3 x [2, -1, 4] = [6, -3, 12]. Adding [1, 1, 1] gives [6 + 1, -3 + 1, 12 + 1] = [7, -2, 13].

AQA A-Level Computer Science 7517

Fundamentals of Programming

Topic overview Data types Programming concepts and program structure Arithmetic operations Relational Operations, Constants and Variables Boolean operations String-handling operations Random numbers and exception handling Subroutines, parameters and return values Variable scope and stack frames Recursive techniques Programming paradigms and the procedural approach Classes, Objects, Inheritance and Polymorphism Aggregation, Composition, Class Diagrams and Design Principles

Fundamentals of Data Structures

Topic overview Data Structures and Abstract Data Types Static and Dynamic Data Structures Queues and Linear Queues Circular and Priority Queues Stacks Graphs Trees Hash tables Dictionaries Vectors and Their Notation Vector Arithmetic and Geometry

Fundamentals of Algorithms

Topic overview Graph traversal Tree traversal Reverse Polish notation Searching algorithms Sorting algorithms Optimisation algorithms

Theory of Computation

Topic overview Logic Problems and Algorithm Constructs Pseudo-Code, Hand-Tracing and Program Correctness Representational Abstraction, Generalisation and Information Hiding Procedural and Functional Abstraction Data Abstraction and Problem Reduction Decomposition and Composition Automation and Finite State Machines Sets and Set Notation Subsets and Set Operations Regular expressions and regular languages Backus-Naur Form and syntax diagrams Comparing Algorithms and Functions Order of complexity and Big-O notation Limits of computation, tractability and heuristics Computable problems and the Halting problem Turing machines

Fundamentals of Data Representation

Topic overview Number systems and the sets of numbers Number bases: decimal, binary and hexadecimal Bits, bytes and units of information Unsigned binary and binary arithmetic Signed binary using two's complement Fixed point and floating point representation Normalisation of floating point numbers Rounding errors, accuracy, range and precision Character coding systems Error checking and correction Bit patterns, analogue and digital Bitmapped graphics Vector graphics Digital representation of sound MIDI and Data Compression Encryption

Fundamentals of Computer Systems

Topic overview Hardware, Software and Its Classification System Software and the Operating System Translators and the Role of the Operating System Classification of programming languages Types of program translator Logic Gates and Their Truth Tables Logic Circuits and D-Type Flip-Flops Half-Adders and Full Adders Boolean algebra

Fundamentals of Computer Organisation and Architecture

Topic overview Internal components, buses and von Neumann and Harvard architectures The Stored Program Concept and the Processor The Fetch-Execute cycle The processor instruction set Addressing modes Machine-Code Operations Shifts, Halting and Assembly Language Programs Interrupts Factors affecting processor performance Input and output devices Secondary storage devices

Fundamentals of Communication and Networking

Topic overview Communication methods Communication basics Network Topology and Types of Networking Wireless networking The Internet and Packet Switching Routing, URLs and Domain Names Firewalls, Worms, Trojans and Viruses Encryption, Digital Signatures and Certificates TCP/IP Standard application layer protocols IP Address Structure and Subnet Masking IP Standards and Public and Private Addresses DHCP and Network Address Translation Client server model Thin- versus thick-client computing

Consequences of Uses of Computing

Topic overview The Four Kinds of Issue in Computing Legislation, Ethical Dilemmas and Case Studies

Systematic Approach to Problem Solving

Topic overview Analysis, Design and Implementation Testing and Evaluation

Fundamentals of Databases

Topic overview Conceptual data models and entity relationship modelling Relational databases Database design and normalisation techniques Structured Query Language (SQL)Client server databases

Big Data

Topic overview Volume, Velocity and Variety Why Big Data Resists Analysis Distributed Processing and Functional Programming Fact-Based Models and Graph Schemas

Fundamentals of Functional Programming

Topic overview Function Types, First-Class Objects and Function Application Partial Function Application and Composition Writing functional programs Lists in functional programming

Fundamentals of Data Structures

Vector Arithmetic and Geometry

2 things to know
Paper 1

Overview

Recall questions on visualising a vector as an arrow, adding vectors, scalar multiplication and the dot product.

What you need to know

Answer each one before you reveal it.

Your recallNot started

Numbers

Give the formula for the dot product of two vectors.

Key term

Give two applications of the dot product.

Questions from this subtopic

Each one with the model answer underneath.

1
Explain how a vector can be visualised as an arrow.
Model answer
Draw the arrow so that it begins at the origin and finishes at the point whose coordinates are the vector's entries. The 2-vector [2.0, 3.0] over ℝ therefore becomes the arrow running from (0, 0) to (2.0, 3.0). Once drawn this way, the arrow's length carries the vector's magnitude and the way it points carries its direction. A 3-vector is pictured identically, with the arrow reaching from the origin out to a point in three-dimensional space.
2
How are two vectors added?
Model answer
Entry by entry: the entry at each position of the result is the sum of the entries at that position in the two vectors. The vectors must have the same number of entries. For example [2, 3] + [4, 1] = [2 + 4, 3 + 1] = [6, 4].
3
What does vector addition achieve geometrically?
Model answer
Translation. Adding a vector to a point moves that point by the amount the added vector describes, without changing its orientation or size. Adding [4, 1] to the point [2, 3] moves it 4 to the right and 1 up, giving [6, 4]. Drawn as arrows, the sum is found by placing the tail of the second arrow at the head of the first.
4
How is a vector multiplied by a scalar?
Model answer
Every entry of the vector is multiplied by the scalar. For example 2 x [3, -1] = [6, -2], and 0.5 x [4, 8] = [2, 4].
5
What does scalar-vector multiplication achieve geometrically?
Model answer
Scaling. Multiplying by a scalar changes the length of the arrow by that factor while leaving its direction along the same line: a scalar greater than 1 stretches it, a scalar between 0 and 1 shrinks it, and a negative scalar reverses the direction as well as scaling the length by the size of the scalar.
6
Calculate 3 x [2, -1, 4] and then add [1, 1, 1].
Model answer
3 x [2, -1, 4] = [6, -3, 12]. Adding [1, 1, 1] gives [6 + 1, -3 + 1, 12 + 1] = [7, -2, 13].

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