Formula Derivation

[Ubwino Kelvin]

Help please: Could you derive one formula (equation) from this?
t is a time quantity
x and y and z are spatial coordinates.
the four 0s in the matrix on the right are zeroes. Thanks

 

[Deleted member 4993]

Help please: Could you derive one formula (equation) from this?
t is a time quantity
x and y and z are spatial coordinates.
the four 0s in the matrix on the right are zeroes. Thanks
View attachment 28880

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[HallsofIvy]

Do you realize that, subtracting, [imath]t_0[/imath], [imath]x_0[/imath], [imath]y_0[/imath], and [imath]z_0[/imath] all cancel?

This equation is simply [math]\begin{pmatrix}t_1- t_p \\x_1- x_p \\ y_1- y_p \\ z_1- z_p \end{pmatrix}= \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}[/math]
from which we conclude that [math]\begin{pmatrix}t_1 \\ x_1 \\ y_1 \\ z_1\end{pmatrix}= \begin{pmatrix}t_p \\ x_p \\ y_p \\ z_p\end{pmatrix}[/math].[/imath]