Parametric Equation question

[markraz]

Hi, I saw this version of a parametric equation in an example somewhere



I always thought parametric equation looked like this:
x= x1 + (x)t
y= y1 + (y)t
z= z1 + (z)t

why do they represent the length by (x2-x1)t ,(y2-y1)t , (z2-z1)t ??
what's the purpose of the extra subtraction? is that because it is not normalize?


thanks in advance

 

[Dr.Peterson]

Hi, I saw this version of a parametric equation in an example somewhere

View attachment 23680

I always thought parametric equation looked like this:
x= x1 + (x)t
y= y1 + (y)t
z= z1 + (z)t

why do they represent the length by (x2-x1)t ,(y2-y1)t , (z2-z1)t ??
what's the purpose of the extra subtraction? is that because it is not normalize?


thanks in advance

A parametric equation (for a line, in this case) can have any form you like. I don't know what you mean by (x), (y), (z), but if they are supposed to be constants, then that's exactly the same form.

The form you are asking about is designed so that for [MATH]t=0[/MATH], the point is [MATH]P_1 = (x_1, y_1, z_1)[/MATH], while for t=1, the point is [MATH]P_2 = (x_2, y_2, z_2)[/MATH]. The form you are comparing it to uses an initial point [MATH](x_1, y_1, z_1)[/MATH] and a vector ("(x)", "(y)", "(z)") which is in fact the vector [MATH]P_1P_2[/MATH].

Everything makes good sense, especially if you are familiar with vectors. In vector form, either equation is [MATH]\mathbf{X} = \mathbf{P_1}+t\mathbf{V}[/MATH], where [MATH]\mathbf{V} = \mathbf{P_2}-\mathbf{P_1}[/MATH].