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Maths

Vectors - Higher

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

toggle answer

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) .

Back to Geometry and measures index

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