In this vector lesson, we’ll look at position vectors vs direction vector.
Position vector
A position vector tells us the direction a point is relative to a reference point.
In x,y or x,y,z- coordinate system, the reference point refers to the origin where x, y (and z) are 0.
From the diagram below, the position of point A with respect to the origin, O, can be described as:
- 4 units in the positive x-direction,
- 2 units in the positive y direction, and
- 5 units in the position z direction.
Once the position from the reference point changes, the position vector changes. E.g. points A and B are different points and hence are different position vectors.
Direction Vector
A direction vector tells us the orientation of the vector. Any direction vectors that are (i) parallel (ii) pointing in the same direction and (iii) have the same magnitude are equal.
Considering the position vector of A and B:
A direction vector of AB, involves moving from point A to B. To go from point A to B, we’ll:
- move -5 units in the x-direction
- move -2 units in the y-direction
- move -4 units in the z- direction
Hence, when the direction vector of AB can be written as:
Any vector that involves the following displacement will be equal to the direction vector of AB:
- move -5 units in the x-drection
- move -2 units in the y-direction
- move -4 units in the z- direction
In the diagram below, we represent the direction vector of AB with the black arrow. Notice that the blue and red arrows are both (i) parallel (ii) pointing in the same direction and (iii) of the same magnitude to direction vector AB. Hence, all the 3 direction vectors shown below are equal.
Useful Results of Direction Vector
When given 2 position vectors, we can find their direction using this formula:
Direction vector of AB involves finding the displacement from point A (starting point) to B (end point); while direction vector BA involves finding the displacement from point B (starting point) to point A (end point).
Hence, direction vector of AB and BA are in opposite direction or
Notes on H2 Math Vectors
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