3.4 Coordinates

If a person had 2 independent vectors in R2, one could describe the location of any point in R2 as a linear

combination of those vectorsThat combination of vectors could be seen a the “address”

that vector.

We usually use a Cartesian coordinate system and write a location of an object in terms of i and j (and k in 3D)

However, we could describe the location of a point just as easily with any independent vectors.

We could create a coordinate grid as follows

We will explain why we would want to do this later in this lecture

Coordinates in a subspace of Rn

Consider the basis β = (v1,v2,v3…vm)

of a subspace V in Rn

Any Vector can be written uniquely as

x = cv1+cv2 …+cvm

The scalars c1,c2, …cm are called the coordinates of x

β is the coordinate vector denoted by [ x] β. Thus

[ x ] β = means x = cv1+cv2 …+cvm

c1

c2

cm

…

Problem 2

Determine if x is in the span of v1 and v2

If so find x with respect to the basis v1 and v2

Solution to Problem 2

Problem 8

Determine if x is in the span of v1 and v2

If so find x with respect to the basis v1 and v2

Solution to problem 8

Problem 10Determine if x is in the span of v1 and v2

If so find x with respect to the basis v1 and v2

Solution to problem 10

Proceeding as per problem 2 we find

Application of Coordinates

The diagram on the next slide is called a space time diagram.

Each grid creates a separate coordinate system that both show the location in space.

This is a space time diagram. The horizontal axis represents the spatial position (one dimensional) the

vertical axis represents the time dimensionThe black axis represents a Stationary frame. The circle represents simultaneous events from his point of view. All horizontal lines are simultaneous to each other from his point of view.

The red frame is a moving frame. As this person is moving light from an event behind him takes a longer time to reach him and he runs into the light ahead of him more quickly (compared to the stationary person)

The red solid line represents simultaneous events from this moving frames point of view.

Both grids represent time and space coordinates from people in different frames.

It is useful to be able to calculate coordinates and change between frames.

The lines parallel to x’ show lines of events that are simultaneous in a frame moving at ¼ the speed of light.

As people move faster closer to the speed of light, light coming from in front of them reaches them more quickly and light from behind them takes

longer to catch them

The vertical axis is the space axis, the horizontal axis is the time axis

Example from Relativity

If a point at (3,4) in standard coordinates.

Find its coordinates on the axis above pretend that the basis is (it would not be exactly this)

<1/4 , 1> ,<1, 1/4>

Solution to Example from Relativity

a ¼ + b 1 = 3

1 ¼ 4

¼ a +b = 3

a + ¼ b = 4

a = 52/15 b = 32/15 which represents the coordinate vector

Please note that the person in the moving frame gave this a lower time value or perceived it happening before the person in the stationary frame. (If you are moving towards and event you can run into the light faster than if you are standing still. This only makes a difference if you are traveling at speeds that approach the speed of light)

[ ] [ ] [ ]

More information on Space time diagrams check out the book

Very Special Relativity from Mr. Whitehead

Homework: p. 147 1-17 odd