Vector Pathways (SQA National 5 Maths): Revision Note

Vector pathway basics

How do I find the vector between two points?

• A vector pathway is a path of vectors taking you from a start point to an end point

• The following grid is made up entirely of parallelograms

  • The vectors a and b are defined as marked in the diagram:

    • Any vector that goes horizontally to the right along a side of a parallelogram will be equal to a

    • Any vector that goes up diagonally to the right along a side of a parallelogram will be equal to b

• To find the vector between two points

  • Count how many times you need to go horizontally to the right

    • This will tell you how many a's are in your answer

  • Count how many times you need to go up diagonally to the right

    • This will tell you how many b's are in your answer

  • Add the a's and b's together

    • E.g. 

• You will have to put a negative in front of the vector if it goes in the opposite direction

  • -a is one length horizontally to the left

  • -b is one length down diagonally to the left

    • E.g.  or

    • Likewise,

• It is possible to describe any vector that goes from one point to another in the above diagram in terms of a and b

Finding more challenging vector pathways

How can vector pathway questions be made more challenging?

• You may need to find expressions for vectors in places where you cannot simply count spaces on a grid

• You should be familiar with the properties of different types of triangle and quadrilateral

  • You may need to use these properties to answer a vector pathway question

• Look out for places where two vectors in a diagram are equal or where one is a multiple of the other

How do I use multiples in vector pathways?

• When multiplying a vector by a scalar number, you can use the normal rules of algebra

  • E.g. expanding brackets, collecting like terms

• In the example shown, if  and you know that then

  • 

• Questions may specify that a point is the midpoint of a line segment

  • E.g. point may be the midpoint of line segment

  • This means that and

• Note that if one vector is a multiple of another, this means that the vectors are parallel

  • E.g. is parallel to

    • It is twice as long and points in the same direction

  • is also parallel to

    • It is three times as long and points in the opposite direction (because of the minus sign)

• If two vectors are parallel, have the same length and point in the same direction

  • then they are equal