Many concepts concerning vectors in can be extended to other mathematical systems. We can think of a vector space in general, as a collection of objects that behave as vectors do in . The objects of such a set are called vectors.A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), satisfying the following properties: Let u, v, and w be vectors in V, and let c and d be scalars.Addition:
() is in V.
1. .
2. .
3. V has a zero vector 0 such that for every u in V, .
4. For every u in V, there is an element in V, .
Scalar multiplication:
() is in V.
5. .
6. .
7. .
8. .
1.21 PLOTTING POINTS IN THREE DIMENSIONSWe can expand our 2-dimensional (x-y) coordinate system into a 3-dimensional coordinate system, using x-, y-, and z-axes.
The x-y plane is horizontal in our diagram above and shaded green. It can also be described using the equation z = 0, since all points on that plane will have 0 for their z-value.
The x-z plane is vertical and shaded pink above. This plane can be described using the equation y=0.
The y-z plane is also vertical and shaded blue. The y-z plane can be described using the equation x=0.
We normally use the ‘right-hand orientation’ for the 3 axes, with the positive x-axis pointing in the direction of the first finger of our right hand, the positive y-axis pointing in the direction of our second finger and the positive z-axis pointing up in the direction of our thumb.
Example – Points in 3-D Space
In 3-dimensional space, the point (2,3,5) is graphed as follows:To reach the point (2,3,5), we move 2 units along the x-axis, then 3 units in the y-direction, and then up 5 units in the z-direction.
Distance in 3-dimensional Space
To find the distance from one point to another in 3-dimensional space, we just extend Pythagoras’ Theorem.
Distance from…