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 **v **| 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 **v **and **b** and we want to create a set that defines the points on the three axis, that is s={**v **+ t**b** | t ER }. As above, this means, vector **v** added to t multipled by vector** v **minus vector **b,** where t is a real number.

We need to take vector **v** minus vector **b **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.

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