3.4 Coordinates

If a person had 2 independent vectors in R 2, one could describe the location of any point in R 2 as a linear combination of those vectors That 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 lectureWe will explain why we would want to do this later in this lecture

Coordinates in a subspace of R n Consider the basis β = (v 1,v 2,v 3 …v m ) of a subspace V in R n Any Vector can be written uniquely as x = cv 1 +cv 2 …+cv m The scalars c 1,c 2, …c m are called the coordinates of x β is the coordinate vector denoted by [ x] β. Thus [ x ] β = means x = cv 1 +cv 2 …+cv m c1c2cmc1c2cm …

Problem 2 Determine if x is in the span of v 1 and v 2 If so find x with respect to the basis v 1 and v 2

Solution to Problem 2

Problem 8 Determine if x is in the span of v 1 and v 2 If so find x with respect to the basis v 1 and v 2

Solution to problem 8

Problem 10 Determine if x is in the span of v 1 and v 2 If so find x with respect to the basis v 1 and v 2

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 dimension The 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),

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 odd