Parametric Vectors

In the below, we have our vector (2,1). As we know, if we multiply that vector by a scalar unit, it will change the magnitude (length) of the vector, but not the direction. This is great, but, we need to create a ‘set’ which can define the entire line that would be created by multiplying our vector by many numbers.

For example, by multipying v by the below values, we are creating the co-ordinates for a line on the graph. We represent this as shown at the bottom of the below image s = { c | cER }. This means, the set is equal to c, multiplied by vector v, where c is a real number. 

  • v * 2 = (4,2)
  • v * 3 = (6, 3)
  • v * 4 = (8, 4)

If we wanted the same line to cross through the point (1,2) which is the vector of x we would add vector x to each point in vector v.

Now we have something a little harder to visualise. A 3 dimensional vector. In the below, I have shown where the X, Y and Z axis may be on the chart. We have vectors and b and we want to create a set that defines the points on the three axis, that is s={+ tb | t ER }. As above, this means, vector v added to t multipled by vectorminus vector b, where t is a real number.

We need to take vector v minus vector and then fill the values into the formula. From here, we can see that:

  • X is going to equal (-1, -1t)
  • Y will be (2, -1t)
  • Z will be (7,3t)

Where t is a variable scalar we use to derive the line.