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Vectors (AQA GCSE Maths)

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  • What is a vector?

    A vector is a type of number with both size and direction.

    A vector with its magnitude and direction labelled.

  • True or False?

    stack B A with rightwards arrow on top represents the vector from B to A.

    True.

    stack B A with rightwards arrow on top represents the vector from B to A.

  • How do you write "3 to the left and 2 up" as a column vector?

    "3 to the left and 2 up" can be written as the column vector open parentheses table row cell negative 3 end cell row 2 end table close parentheses.

  • True or False?

    The vector open parentheses table row 2 row 3 end table close parentheses must be drawn starting at open parentheses 0 comma space 0 close parentheses and ending at the point open parentheses 2 comma space 3 close parentheses.

    False.

    The vector open parentheses table row 2 row 3 end table close parentheses does not need to be drawn starting at open parentheses 0 comma space 0 close parentheses and ending at the point open parentheses 2 comma space 3 close parentheses.

    The vector can start at any point, as long as it goes 2 to the right and 3 up.

  • How do you add two column vectors?

    To add two column vectors, add together the numbers in the corresponding parts of the vector.

    For example, open parentheses table row 1 row 2 end table close parentheses plus open parentheses table row 3 row cell negative 5 end cell end table close parentheses equals open parentheses table row cell 1 plus 3 end cell row cell 2 plus open parentheses negative 5 close parentheses end cell end table close parentheses equals open parentheses table row 4 row cell negative 3 end cell end table close parentheses.

  • True or False?

    5 cross times open parentheses table row 3 row 4 end table close parentheses equals open parentheses table row 15 row 4 end table close parentheses

    False.

    5 cross times open parentheses table row 3 row 4 end table close parentheses not equal to open parentheses table row 15 row 4 end table close parentheses.

    To multiply a vector by a scalar, you need to multiply all parts of the vector by the scalar.

    5 cross times open parentheses table row 3 row 4 end table close parentheses equals open parentheses table row cell 5 cross times 3 end cell row cell 5 cross times 4 end cell end table close parentheses equals open parentheses table row 15 row 20 end table close parentheses.

  • If you know what the vectors bold a and bold b look like, how can you draw the vector bold a plus bold b?

    If you know what the vectors bold a and bold b look like, to draw the vector bold a plus bold b:

    • Draw the vector bold a

    • Draw the vector bold b starting at the end of bold a

    • Draw a vector from the start of bold a to the end of bold b

    An arrow representing vector a has an arrow representing vector b starting from its end point. The resultant vector, a + b is drawn as an arrow that starts from the start point of vector a and ends at the end point of vector b.

  • True or False?

    The vector 2 bold a is in the same direction as bold a but is double the length of bold a.

    True.

    The vector 2 bold a is in the same direction as bold a but is double the length.

    Scalar multiples of a vector change the length of the vector.

  • What is the geometrical relationship between the vectors bold a and negative bold a?

    The vectors bold a and negative bold a have the same length but they are in opposite directions.

  • What mathematical theorem can be applied to find the length of a vector?

    Pythagoras' theorem can be applied to find the length of a vector .

  • True or False?

    The length of the vector open parentheses table row 2 row 3 end table close parentheses is 2 squared plus 3 squared.

    False.

    The length of the vector open parentheses table row 2 row 3 end table close parentheses is not 2 squared plus 3 squared.

    To find the magnitude of a vector, you square the numbers, add them together and then take the positive square root (using Pythagoras).

    The length of the vector open parentheses table row 2 row 3 end table close parentheses is square root of 2 squared plus 3 squared end root.

  • True or False?

    The length of the vector open parentheses table row 2 row cell negative 3 end cell end table close parentheses is square root of 2 squared minus 3 squared end root.

    False.

    The length of the vector open parentheses table row 2 row cell negative 3 end cell end table close parentheses is not square root of 2 squared minus 3 squared end root.

    You need to square the negative in front of the 3.

    The length of the vector open parentheses table row 2 row cell negative 3 end cell end table close parentheses is square root of 2 squared plus open parentheses negative 3 close parentheses squared end root.

  • True or False?

    stack A B with rightwards arrow on top equals stack O A with rightwards arrow on top minus stack O B with rightwards arrow on top

    False.

    stack A B with rightwards arrow on top not equal to stack O A with rightwards arrow on top minus stack O B with rightwards arrow on top

    The correct equation is stack A B with rightwards arrow on top equals stack O B with rightwards arrow on top minus stack O A with rightwards arrow on top.

  • True or False?

    stack A B with rightwards arrow on top plus stack B C with rightwards arrow on top equals stack A C with rightwards arrow on top

    True.

    stack A B with rightwards arrow on top plus stack B C with rightwards arrow on top equals stack A C with rightwards arrow on top

    This means if you start at A then go to B and then go to C, this is the same vector as starting at A and going to C.

  • How else could you write the vector negative stack A B with rightwards arrow on top?

    The vector negative stack A B with rightwards arrow on top can also be written as stack B A with rightwards arrow on top.

  • True or False?

    If two vectors are parallel, then they are scalar multiples.

    True.

    If two vectors are parallel, then they are scalar multiples.

    E.g. the vectors 3 bold a and 12 bold a are parallel.

  • True or False?

    If the point P divides the line AB in the ratio 1 : 3, then stack A P with rightwards arrow on top equals 1 third stack A B with rightwards arrow on top.

    False.

    If the point P divides the line AB in the ratio 1 : 3, then stack A P with rightwards arrow on top not equal to 1 third stack A B with rightwards arrow on top.

    P divides the line into 4 equal parts, with 1 part on one side, and 3 parts on the other side.

    The correct equation is stack A P with rightwards arrow on top equals 1 fourth stack A B with rightwards arrow on top.

  • If the vectors stack A B with rightwards arrow on top and stack A C with rightwards arrow on top are parallel, what does this tell you about the three points, A, B and C?

    If the vectors stack A B with rightwards arrow on top and stack A C with rightwards arrow on top are parallel, then the three points A, B and C all lie on the same straight line.

  • Points A and B have position vectors bold a and bold b, respectively.

    Write a formula for the position vector of the midpoint of A and B in terms of bold a and bold b.

    Points A and B have position vectors bold a and bold b, respectively.

    A formula for the position vector of the midpoint of A and B is 1 half open parentheses bold a plus bold b close parentheses.

Previous:3D Pythagoras & TrigonometryNext:Transformations

    1. Topic 1

      Number Operations

    2. Topic 2

      Prime Factors, HCF & LCM

    3. Topic 3

      Powers, Roots & Standard Form

    4. Topic 4

      Fractions

    5. Topic 5

      Percentages

    6. Topic 6

      Simple & Compound Interest, Growth & Decay

    7. Topic 7

      Fractions, Decimals & Percentages

    8. Topic 8

      Rounding, Estimation & Bounds

    9. Topic 9

      Surds

    10. Topic 10

      Using a Calculator

    1. Topic 1

      Introduction to Algebra

    2. Topic 2

      Algebraic Roots & Indices

    3. Topic 3

      Expanding Brackets

    4. Topic 4

      Factorising

    5. Topic 5

      Completing the Square

    6. Topic 6

      Algebraic Fractions

    7. Topic 7

      Rearranging Formulas

    8. Topic 8

      Algebraic Proof

    9. Topic 9

      Linear Equations

    10. Topic 10

      Quadratic Equations

    11. Topic 11

      Simultaneous Equations

    12. Topic 12

      Iteration

    13. Topic 13

      Forming & Solving Equations

    14. Topic 14

      Functions

    15. Topic 15

      Coordinate Geometry

    16. Topic 16

      Linear Graphs

    17. Topic 17

      Graphs of Functions

    18. Topic 18

      Equation of a Circle

    19. Topic 19

      Estimating Gradients & Areas under Graphs

    20. Topic 20

      Real-Life Graphs

    21. Topic 21

      Solving Inequalities

    22. Topic 22

      Graphing Inequalities

    23. Topic 23

      Transformations of Graphs

    24. Topic 24

      Sequences

    1. Topic 1

      Ratios

    2. Topic 2

      Problem Solving with Ratios

    3. Topic 3

      Direct & Inverse Proportion

    4. Topic 4

      Standard & Compound Units

    5. Topic 5

      Exchange Rates & Best Buys

    1. Topic 1

      Symmetry & Shapes

    2. Topic 2

      Angles in Polygons & Parallel Lines

    3. Topic 3

      Bearings, Scale Drawing, Constructions & Loci

    4. Topic 4

      Circle Theorems

    5. Topic 5

      Area & Perimeter

    6. Topic 6

      Circles, Arcs & Sectors

    7. Topic 7

      Volume & Surface Area

    8. Topic 8

      Congruence, Similarity & Geometrical Proof

    9. Topic 9

      Area & Volume of Similar Shapes

    10. Topic 10

      Pythagoras & Trigonometry

    11. Topic 11

      Sine, Cosine Rule & Area of Triangles

    12. Topic 12

      3D Pythagoras & Trigonometry

    13. Topic 13

      Vectors

    14. Topic 14

      Transformations

    1. Topic 1

      Introduction to Probability

    2. Topic 2

      Simple Probability Diagrams

    3. Topic 3

      Tree Diagrams

    4. Topic 4

      Combined & Conditional Probability

    1. Topic 1

      Averages, Ranges & Data

    2. Topic 2

      Statistical Diagrams

    3. Topic 3

      Histograms

    4. Topic 4

      Cumulative Frequency & Box Plots

    5. Topic 5

      Scatter Graphs & Correlation