# Change of basis (coordinates)

can somebody help me with this by giving me a hint how to solve these $2$ $?$:

Given

• a point $P$ with coordinates $(10,4)$ in a fixed coordinate system.

• a new coordinate system with origo in $(6,0)$ and rotated $60^{\circ}$ counterclockwise.

$1$) Find the coordinates of $P$ in the new coordinate system.

$2$) Write down the equation, that will transform any point $P=(x,y)$ in the fixed coordinate system to coordinates $P'=(x’,y’)$ in the new coordinate system.

This is what I have thought about: Finding the coordinates as projections and then I don't know because I don't understand what I have. Do I've $2$ coordinates$?$

• The value of the x coordinate of P in the old system is 10. What is its value in the new coordinate system? And that of y?
– MASL
Commented Sep 22, 2015 at 18:21
• Consider now another arbitrary point in that in the old system has coordinates (2,3). What are its x and y values in the new system?
– MASL
Commented Sep 22, 2015 at 18:24
• Finally, consider an arbitrary point $(x_o,y_o)$ in the old system. Can generalize from the above examples and write the values of (x,y) in the new system in terms of $x_o,\,y_o$?
– MASL
Commented Sep 22, 2015 at 18:25
• Do I have to say 10-2=x and 4-3=y ? Commented Sep 22, 2015 at 18:25
• No. Ok, let's start with something easier. Consider the real line, from $-\infty$ to the left to $+\infty$ to the right, and a point $P$ at $x=10$. No suppose I move the origin of coordinate and locate it where before was $6$. What's the location of $P$ now?
– MASL
Commented Sep 22, 2015 at 18:30

When origin located at $O=(0,0)$: $P=(x_o,y_o)$

Location of new origin: $O'=(a,b)$

Point $P$ in new system: $P=(x_o,y_o)-(a,b)=(x_o-a,y_o-b)$

• However, if you don't try to solve and understand the questions I asked you before, no formula will really be helpful. You need to understand that on a paper by drawing it all!!
– MASL
Commented Sep 22, 2015 at 21:30
• the new x is 4 as 10-6=4 Commented Sep 22, 2015 at 23:09
• You got it. If this helped you it would be nice if you accept and/or upvote this answer. Thanks.
– MASL
Commented Nov 20, 2015 at 14:45