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Flashcards · Maths

Vectors, Constructions and Loci

80 cards from 5 lessons

  1. What is a vector?

    A quantity with a size and a direction, such as a movement.

  2. What does the top number of a column vector show?

    The horizontal move, right if positive.

  3. What does the bottom number of a column vector show?

    The vertical move, up if positive.

  4. What does \(\overrightarrow{AB}\) mean?

    The vector from \(A\) to \(B\).

  5. What is \(\overrightarrow{BA}\) in terms of \(\overrightarrow{AB}\)?

    \(-\overrightarrow{AB}\).

  6. How do you find a vector from two coordinates?

    Subtract the coordinates of the start from the end.

  7. How do you add column vectors?

    Add the top numbers and add the bottom numbers.

  8. How do you subtract column vectors?

    Subtract the top numbers and subtract the bottom numbers.

  9. How do you multiply a vector by a number?

    Multiply both numbers in the column by it.

  10. When are two vectors equal?

    When they have the same size and direction.

  11. When are two vectors parallel?

    When one is a multiple of the other.

  12. What does a negative multiple of a vector do?

    It points in the opposite direction.

  13. How do you draw \(\mathbf{a} + \mathbf{b}\)?

    Draw \(\mathbf{b}\) starting from the end of \(\mathbf{a}\), then join the start to the finish.

  14. How do you draw \(\mathbf{a} - \mathbf{b}\)?

    Add \(\mathbf{a}\) and \(-\mathbf{b}\).

  15. How do you find the length of a column vector?

    Use Pythagoras' theorem on the two numbers (Higher tier).

  16. What is the length of \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\)?

    5 (Higher tier).

  17. If \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\), what is \(\overrightarrow{AB}\)?

    \(\mathbf{b} - \mathbf{a}\).

  18. What is \(\overrightarrow{AO}\) in terms of \(\overrightarrow{OA}\)?

    \(-\overrightarrow{OA}\).

  19. How do you find the vector to the midpoint of \(AB\)?

    \(\overrightarrow{OM} = \overrightarrow{OA} + \dfrac{1}{2}\overrightarrow{AB}\), which is \(\dfrac{1}{2}(\mathbf{a} + \mathbf{b})\).

  20. What fraction of \(AB\) is \(AP\) when \(AP : PB = 3 : 1\)?

    \(\dfrac{3}{4}\).

  21. How do you find the vector to a point on \(AB\)?

    Go to \(A\), then add the fraction of \(\overrightarrow{AB}\) to reach the point.

  22. When are two vectors parallel?

    When one is a multiple of the other.

  23. How do you prove lines are parallel?

    Show that their vectors are multiples of each other.

  24. When are three points collinear?

    When two of the vectors between them are parallel and share a point.

  25. What do you write at the end of a collinear proof?

    The vectors are parallel and share a point, so the points are on a straight line.

  26. What does \(\overrightarrow{AB} = 2\overrightarrow{BC}\) show?

    \(A\), \(B\) and \(C\) are on a straight line, with \(AB\) twice as long as \(BC\).

  27. How do you find the length of a column vector?

    \(\sqrt{x^2 + y^2}\).

  28. What is the length of \(\begin{pmatrix} 5 \\ 12 \end{pmatrix}\)?

    13.

  29. Why choose a route using known vectors?

    You can add them to reach any point.

  30. How should you simplify a vector expression?

    Collect the terms in each letter.

  31. What is the opposite of \(\mathbf{a} - \mathbf{b}\)?

    \(\mathbf{b} - \mathbf{a}\).

  32. Why draw a diagram in a vector question?

    It shows which vectors are known and which route to use.

  33. What is a construction?

    A drawing made with a ruler and compasses, with no protractor.

  34. What is the perpendicular bisector of a line?

    A line that cuts it in half at right angles.

  35. How do you construct a perpendicular bisector?

    Draw matching arcs from both ends that cross twice, and join the crossing points.

  36. What is true of every point on a perpendicular bisector?

    It is the same distance from both ends of the line.

  37. How do you bisect an angle?

    Draw an arc on both arms, then matching arcs from those points, and join the vertex to the crossing.

  38. What is true of every point on an angle bisector?

    It is the same distance from both arms.

  39. How do you construct a \(60^\circ\) angle?

    Draw two arcs of the same radius and join the end to the crossing point.

  40. How do you construct a \(30^\circ\) angle?

    Construct a \(60^\circ\) angle and bisect it.

  41. How do you construct a triangle from three sides?

    Draw one side, then arcs of the other two lengths from its ends.

  42. Why must you leave the arcs on the drawing?

    They show the method, and earn the marks.

  43. How do you construct a perpendicular from a point to a line?

    Arc from the point cutting the line twice, then arcs from those points, and a line through the crossing.

  44. What is the shortest distance from a point to a line?

    The perpendicular distance.

  45. What instrument do you need for constructions?

    A pair of compasses and a ruler.

  46. Should the compass width stay the same for matching arcs?

    Yes.

  47. What accuracy is usually allowed?

    About 2 millimetres.

  48. What is a bisector?

    A line that cuts something into two equal parts.

  49. What is a locus?

    The set of all points that follow a rule.

  50. What is the plural of locus?

    Loci.

  51. What is the locus of points a fixed distance from a point?

    A circle with that point as centre.

  52. What is the locus of points a fixed distance from a line?

    Parallel lines on both sides, with rounded ends.

  53. What is the locus of points equidistant from two points?

    The perpendicular bisector of the line joining them.

  54. What is the locus of points equidistant from two lines?

    The bisector of the angle between the lines.

  55. What does "less than 3 cm from \(P\)" mean?

    The region inside the circle of radius 3 cm around \(P\).

  56. What does "closer to \(A\) than to \(B\)" mean?

    The side of the perpendicular bisector that contains \(A\).

  57. What does "within 5 m of a wall" mean?

    Inside the parallel line drawn 5 m from the wall.

  58. How do you find a region with several rules?

    Draw each locus and shade the area where all the rules are true.

  59. Why use a scale drawing?

    To fit a real situation on the paper.

  60. What should you leave on a locus drawing?

    The construction arcs.

  61. Which instruments are needed?

    A ruler and a pair of compasses.

  62. How accurate should a locus drawing be?

    Within about 2 mm.

  63. What is the boundary of a region called?

    The locus or edge that separates the shaded part.

  64. What does "equidistant" mean?

    The same distance from two or more things.

  65. What is a bearing?

    A direction given as an angle clockwise from north.

  66. How many figures does a bearing have?

    Three, such as 045.

  67. What is the bearing of east?

    090.

  68. What is the bearing of south?

    180.

  69. What is the bearing of west?

    270.

  70. Where do you measure a bearing from?

    The north line at the starting point.

  71. In which direction do you measure a bearing?

    Clockwise.

  72. How do you find a back bearing when the bearing is less than 180?

    Add 180.

  73. How do you find a back bearing when the bearing is more than 180?

    Subtract 180.

  74. Why are the north lines parallel?

    They all point in the same direction, north.

  75. What does a scale of 1 : 50 000 mean?

    1 cm on the map is 50 000 cm on the ground.

  76. How do you find a real length from a scale drawing?

    Multiply the drawing length by the scale.

  77. What is a plan?

    The view of a solid from above.

  78. What is an elevation?

    The view of a solid from the front or the side.

  79. How are hidden edges shown?

    As dashed lines.

  80. What should you draw at the start of a bearing problem?

    The north line at the start point.