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Vectors (SQA National 5 Maths): Flashcards

Exam code: X847 75

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  • Define a vector.

    A vector is a quantity that has both a size and a direction.

    That is what separates it from a scalar, which is a plain number with no direction attached.

  • What are the three ways of writing a vector?

    A vector may be written as a bold lower-case letter such as \mathbf{a}, underlined when written by hand, or as two capitals with an arrow on top such as \overrightarrow{AB}.

    The third way is component form, which gives the precise distance travelled in each direction.

  • True or False?

    \overrightarrow{AB} and \overrightarrow{BA} are the same vector.

    False.

    The two vectors have the same size but opposite directions, since one runs from A to B and the other from B to A.

    In component form every component simply changes sign.

  • What does the component vector open parentheses table row 6 row 3 end table close parentheses mean?

    The vector means 6 units in the positive x direction and 3 units in the positive y direction.

    So it describes how to get from one point to another, six to the right and three up.

  • How do you find the vector between two points in component form?

    Subtract the coordinates of the first point from the coordinates of the second.

    For A \left(2 , - 3\right) and B \left(7 , 4\right) that gives stack A B with rightwards arrow on top equals open parentheses table row 5 row 7 end table close parentheses.

  • How is the third axis arranged in a set of 3D coordinate axes?

    The z axis is perpendicular to both the x and y axes, so if those are drawn on a sheet of paper it points straight out of it.

    A coordinate \left(x , y , z\right) then gives the distance from the origin along each of the three axes in turn.

  • What does the vector open parentheses table row 2 row cell negative 3 end cell row 4 end table close parentheses mean?

    The vector means 2 units in the positive x direction, 3 units in the negative y direction and 4 units in the positive z direction.

    A negative component always means travelling backwards along that axis.

  • For P \left(3 , 4 , - 2\right) and Q \left(- 1 , 6 , - 1\right) complete the vector \overrightarrow{PQ} in component form.

    stack P Q with rightwards arrow on top equals open parentheses table row cell _ _ _ _ _ _ end cell row 2 row cell _ _ _ _ _ _ end cell end table close parentheses

    The completed vector is:

    stack P Q with rightwards arrow on top equals open parentheses table row cell negative 4 end cell row 2 row 1 end table close parentheses

    Each component is the second point's coordinate minus the first, so the last one is - 1 - \left(- 2\right) = 1.

  • How do you add two vectors in component form?

    Add the matching components separately, so the x components together and the y components together.

    That gives open parentheses table row 5 row 2 end table close parentheses plus open parentheses table row 3 row cell negative 1 end cell end table close parentheses equals open parentheses table row 8 row 1 end table close parentheses.

  • True or False?

    Subtracting one vector from another can give a component larger than either of the originals.

    True.

    Subtracting a negative component adds to the result, so the answer can come out bigger than either starting value.

    In open parentheses table row 2 row cell negative 3 end cell row 5 end table close parentheses minus open parentheses table row 1 row 1 row cell negative 4 end cell end table close parentheses the last component becomes 5 - \left(- 4\right) = 9.

  • What changes when the vectors have three components rather than two?

    Nothing changes in the method, since there is simply a third row to add, subtract or multiply.

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

  • Define a scalar.

    A scalar is an ordinary number, with a size but no direction.

    Multiplying a vector by a scalar multiplies every one of its components by that number.

  • Complete the result of multiplying this vector by a scalar.

    4 open parentheses table row 3 row 2 row cell negative 5 end cell end table close parentheses equals open parentheses table row 12 row cell _ _ _ _ _ _ end cell row cell _ _ _ _ _ _ end cell end table close parentheses

    The completed result is:

    4 open parentheses table row 3 row 2 row cell negative 5 end cell end table close parentheses equals open parentheses table row 12 row 8 row cell negative 20 end cell end table close parentheses

    Every component is multiplied, and a negative one stays negative when the scalar is positive.

  • How do you work out an expression such as 2 \mathbf{a} + 5 \mathbf{b} in component form?

    Multiply each vector by its own scalar first, and only then add or subtract the results.

    For bold a equals open parentheses table row 5 row 2 end table close parentheses and bold b equals open parentheses table row 3 row cell negative 1 end cell end table close parentheses that gives open parentheses table row 10 row 4 end table close parentheses plus open parentheses table row 15 row cell negative 5 end cell end table close parentheses equals open parentheses table row 25 row cell negative 1 end cell end table close parentheses.

  • Define a resultant vector.

    A resultant vector is the single vector an expression simplifies to once all the arithmetic has been done.

    So a vector such as open parentheses table row cell negative 17 end cell row 29 row cell negative 15 end cell end table close parentheses is the resultant of any expressions that produced it.

  • How can an unknown in a vector equation be found?

    Work each side into a single vector, then set the matching components equal to one another.

    From open parentheses table row cell 2 p minus 6 end cell row 9 end table close parentheses equals open parentheses table row 4 row q end table close parentheses the top row gives 2 p - 6 = 4, so p = 5, and the bottom row gives q = 9.

  • Define the magnitude of a vector.

    The magnitude of a vector is its length, also called its modulus.

    It is always positive, because a length cannot be negative and the direction is irrelevant.

  • How is the magnitude of a vector written?

    With vertical bars around the vector, so the magnitude of \mathbf{a} is written \left|\mathbf{a}\right|.

    For a vector named by its endpoints it is written \left|\overrightarrow{AB}\right| instead.

  • Complete the two magnitude formulae by filling in the missing expressions.

    \text{in 2D } \left|\mathbf{a}\right| = \sqrt{\_\_\_\_\_\_}

    \text{in 3D } \left|\mathbf{a}\right| = \sqrt{\_\_\_\_\_\_}

    The completed formulae are:

    \left|\mathbf{a}\right| = \sqrt{x^{2} + y^{2}}

    \left|\mathbf{a}\right| = \sqrt{x^{2} + y^{2} + z^{2}}

    Neither is given on the Formulae List, so both have to be remembered.

  • True or False?

    A vector with negative components has a negative magnitude.

    False.

    Squaring each component removes its sign before the root is taken, so a magnitude is always positive.

    The vector open parentheses table row cell negative 36 end cell row 8 row 24 end table close parentheses has magnitude \sqrt{1296 + 64 + 576} = 44.

  • Why is the magnitude formula the same as Pythagoras' theorem?

    The components of a vector are the two perpendicular sides of a right-angled triangle, and the vector itself is the hypotenuse.

    In three dimensions it matches 3D Pythagoras for exactly the same reason, with a third perpendicular direction added.

  • A vector joins A \left(- 3 , 5\right) to B \left(7 , 1\right) so what is its magnitude?

    First find the components, stack A B with rightwards arrow on top equals open parentheses table row 10 row cell negative 4 end cell end table close parentheses, and then apply the formula.

    That gives \sqrt{10^{2} + \left(- 4\right)^{2}} = \sqrt{116}, which simplifies to 2 \sqrt{29}.

  • What does the magnitude of a vector represent in a real situation?

    The meaning depends on what the vector measures, so the magnitude of a velocity vector is a speed.

    The magnitude of a force vector is the strength of that force, measured in newtons.

  • Define a directed line segment.

    A directed line segment is a line drawn to represent a vector, with its length showing the size.

    An arrowhead, drawn at the end or in the middle, shows the direction.

  • True or False?

    A vector can be drawn anywhere on a grid, as long as its length and direction are right.

    True.

    A vector records only a size and a direction, not a position, so where the arrow starts makes no difference.

    Two arrows of the same length pointing the same way represent the same vector wherever they sit on the grid.

  • How do you draw the vector open parentheses table row 3 row 4 end table close parentheses on a grid?

    Mark any starting point, count 3 units right and 4 units up, and mark a second point there.

    Join the two with a line and put an arrowhead on it pointing at the second point.

  • How do you draw a vector with a negative or a zero component?

    A negative component means counting in the opposite direction, so open parentheses table row 2 row cell negative 4 end cell end table close parentheses goes 2 right and 4 down.

    A zero component means no movement at all in that direction, so open parentheses table row 2 row 0 end table close parentheses is purely horizontal.

  • What does multiplying a vector by a positive scalar do to its arrow?

    The direction stays exactly the same, and the length is multiplied by that scalar.

    So 2 \mathbf{a} is twice as long as \mathbf{a} and points the same way, while \frac{1}{2} \mathbf{a} is half as long.

  • What does multiplying a vector by a negative scalar do to its arrow?

    The direction is reversed, and the length is multiplied by the number after the minus sign.

    So - \mathbf{a} is the same length as \mathbf{a} but points the opposite way, while - 2 \mathbf{a} is twice as long and reversed.

  • How do you draw the sum and the difference of two vectors?

    Draw \mathbf{a}, then draw \mathbf{b} starting where \mathbf{a} ends, and \mathbf{a} + \mathbf{b} runs from the start of \mathbf{a} to the end of \mathbf{b}.

    For \mathbf{a} - \mathbf{b} do the same, but draw - \mathbf{b} second so that its arrow is reversed.

  • Define a vector pathway.

    A vector pathway is a route of vectors joined nose to tail from a start point to an end point.

    Adding the vectors along the route gives the single vector from the start to the end.

  • On a grid of parallelograms how is a vector written using \mathbf{a} and \mathbf{b} together?

    Count how many steps the route takes along each of the two directions, and use those counts as the coefficients.

    Four steps along \mathbf{a} gives 4 \mathbf{a}, and two along \mathbf{b} then three along \mathbf{a} gives 2 \mathbf{b} + 3 \mathbf{a}.

  • What happens when a step in the route runs against the direction of \mathbf{a} itself?

    That step counts as - \mathbf{a}, so four steps backwards give - 4 \mathbf{a}.

    Reversing a whole pathway changes every sign, which is why \overrightarrow{BF} = - \overrightarrow{FB}.

  • True or False?

    There is more than one correct pathway between two points, and they all simplify to the same answer.

    True.

    Any route joining the two points gives the same overall vector, because only the start and the end count.

    Different routes can look very unlike each other before they are simplified, so the answer must always be fully collected.

  • Complete the two halves when M is the midpoint of AB.

    \overrightarrow{AM} = \_\_\_\_\_\_ \overrightarrow{AB}

    \overrightarrow{MB} = \_\_\_\_\_\_ \overrightarrow{AB}

    The completed halves are:

    \overrightarrow{AM} = \frac{1}{2} \overrightarrow{AB}

    \overrightarrow{MB} = \frac{1}{2} \overrightarrow{AB}

    A point a fraction of the way along works the same way, so \overrightarrow{AX} = \frac{3}{8} \overrightarrow{AB} if X is three eighths along.

  • What does it mean when one vector is a multiple of another?

    The two are parallel, since multiplying by a scalar changes only the length and possibly the direction.

    So 2 \left(\mathbf{a} - \mathbf{b}\right) and - 3 \left(\mathbf{a} - \mathbf{b}\right) are both parallel to \mathbf{a} - \mathbf{b}, the second pointing the opposite way.

  • When are two vectors equal to each other?

    Two vectors are equal when they are parallel, the same length, and pointing in the same direction.

    That is why opposite sides of a parallelogram give equal vectors, such as \overrightarrow{QP} = \overrightarrow{RS} in parallelogram PQRS.

  • How is \overrightarrow{AB} found when only \overrightarrow{AC} and \overrightarrow{CB} are given to you?

    Take the route through C, so \overrightarrow{AB} = \overrightarrow{AC} + \overrightarrow{CB}, and then add the components.

    With open parentheses table row 8 row cell negative 2 end cell end table close parentheses and open parentheses table row cell negative 1 end cell row 7 end table close parentheses that gives open parentheses table row 7 row 5 end table close parentheses.

  • How do you find missing coordinates of points on a 3D diagram?

    Use the properties of the solid to relate the unknown point to one whose coordinates you already have.

    Parallel faces, equal edges and symmetry are what supply those relationships.

  • True or False?

    Two points joined by an edge parallel to the y axis differ only in their y coordinate.

    True.

    Moving parallel to one axis changes that coordinate alone and leaves the other two untouched.

    If B is \left(6 , 0 , 7\right) and BE is a 10 unit edge parallel to the y axis, then E is \left(6 , 10 , 7\right).

  • What is useful about the two end faces of a prism?

    The two end faces of a prism are identical in shape and size, and are separated by the length of the prism.

    A vertex on one end face therefore has a matching vertex on the other, differing only along that length.

  • How can an isosceles triangle help you find a missing coordinate?

    An isosceles triangle is symmetrical about the perpendicular from its apex, so its two base vertices sit equal distances either side of that line.

    With one base vertex at x = 3 and the apex above x = 6, the other must be at x = 9.

  • Once you have the coordinates how do you write the vector between two points?

    Subtract the first point's coordinates from the second's, component by component.

    For B \left(6 , 0 , 7\right) and D \left(9 , 10 , 0\right) that gives stack B D with rightwards arrow on top equals open parentheses table row 3 row 10 row cell negative 7 end cell end table close parentheses.

  • Which solids should you know the properties of for these questions?

    Spheres, cones and pyramids from this course, together with cubes, cuboids, prisms and cylinders from National 4.

    Knowing which faces are parallel and which edges are equal is what turns a diagram into coordinates.

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