Maths

Vectors - Higher

A vector describes a movement from one point to another.

Vector notation

A vector quantity has both direction and magnitude (size).

(In contrast a scalar quantity has magnitude only - eg, the numbers 1, 2, 3, 4...)

image: a grid with a diagonal line marked A and B at each respective end. There's an arror in the centre pointing in the upward direction labelled a.

For example this arrow represents a vector. The direction is given by the arrow, while the length of the line represents the magnitude.

This vector can be written as: AB (arrow above) , a, or 3 over 4 .

In print, a is written in bold type. In handwriting, the vector is indicated by putting a squiggle underneath the letter: the letter with a squiggle underneath it indicating a vector

Question

Write down the 3 ways to describe the vector if the arrow is now pointing from B to A.

image: a grid with a diagonal line marked A and B at each respective end. There's an arror in the centre pointing in the downward direction

Answer

Remember that the arrow describes the direction. So, in this case, the vector is from B to A. If we move 'backwards' along a vector, it becomes negative, so a becomes -a. Moving from B to A entails moving 3 units to the left, and 4 down.

So the three ways to write this vector are:

BA (arrow above) , -a and -3 over -4

image: two parallel lines at a diagonal angle, on the left line an arrow points in an upward direction and is labelled a, on the right line an arrow points in the downward direction and is labelled minus a

Vector 'arithmetic'

Equal vectors

If two vectors have the same magnitude and direction, then they are equal.

image: two parallel lines, both are diagonal with arrows marking the an upward direction

Adding vectors

Look at the graph below to see the movements between PQ, QR and PR.

(a over b) + (c  over d) = (a + c over b + d)

Vector PQ (arrow above) followed by vector QR (arrow above) represents a movement from P to R. PQ (arrow above) + QR (arrow above) = PR (arrow above)

Written out the vector addition looks like this

(2 over 5) +  (4 over -3) = (6 over 2)

image: a grid with the points P, Q and R marked. From P to Q the direction of the line is upward, from Q to R the direction is downward

Subtracting vectors

Subtracting a vector is the same as adding a negative version of the vector (remember that making a vector negative means reversing its direction).

(a over b) - (c over b) = (a - c over b - d)

image: a grid with points X, Y and Z joined. The line between X and Y has an arrow indicating an upward direction, the line from Y to Z also had an arrow indicating an upward direction.

Look at the diagram and imagine going from X to Z. How would you write the path in vectors using only the vectors XY (arrow above) and ?

You could say it is vector XY (arrow above) followed by a backwards movement along .

So we can write the path from X to Z as

XY (arrow above) - ZY (arrow above) = XZ (arrow above)

Written out in numbers it looks like this:

(4 over 2) - (1 over 2) = (3 over 0)

Question

If x = 1 over 3 , y = -2 over 4 and z = -1 over -2 , find:

  1. -y
  2. x - y
  3. 2x + 3z

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Answer
  1. (Did you remember to change the signs?)

  2. (1 over 3) - (2 over -4) = (1 - -2 over 3 - 4) = (3 over -4)

  3. 2 (1 over 3) + 3 (-1 over -2) = (2 over 6) + (-3 over -4) = (-1 over 0)

Resultant vectors

To travel from X to Z, it is possible to move along vector XY (arrow above) followed by YZ (arrow above). It is also possible to go directly along XZ (arrow above).

XZ (arrow above) is therefore known as the resultant of XY (arrow above) and YZ (arrow above) .

Geometric problems

Question

Write as single vectors:

  1. f + g
  2. a + b
  3. e - b - a

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Answer
  1. e
  2. -c (Did you remember the minus sign?)
  3. -d

Remember

Two vectors are equal if they have the same magnitude and direction, regardless of where they are on the page. You need to use this fact to answer the next question.

Question

Triangles ABC and XYZ are equilateral.

X is the midpoint of AB, Y is the midpoint of BC, Z is the midpoint of AC.

AX (arrow above) = a , XZ (arrow above) = b , AZ (arrow above) = c

Express each of the following in terms of a, b and c.

  1. XY (arrow above)

  2. YZ (arrow above)

  3. XC (arrow above)

  4. BZ (arrow above)

  5. AC (arrow above)

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Answer
  1. c
  2. -aRemember that YZ (arrow above) is parallel to AX (arrow above) and of the same length, but the direction is different.

  3. b + c

    (It is also possible to move from X to A, and then on to C. This would give the answer -a + 2c. How many other answers can you think of?)

  4. b - a

    Or 2b - c,

    or -2a + c

  5. 2c

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