Hi,

Let's say we have four points: \vec{p_1} = \vec{o} + w\vec{d}, \vec{p_2} = \vec{o} + x\vec{d}, \vec{p_3} = \vec{o} + y\vec{d}, \vec{p_4} = \vec{o} + z\vec{d}
And w \leq x \leq y \leq z

I would like to find \|x\|\vec{d} + \|y\|\vec{d} and compare it with \|\vec{p_4} - \vec{p_1}\|, however, I do not know the values of the variables that make up the points.

An example in 1-D:
\vec{o} = 0,\ \vec{d} = 1,\ w = -3,\ x = -1,\ y = 4,\ z = 5
\vec{p_1} =-3,\ \vec{p_2} = -1,\ \vec{p_3} = 4,\ \vec{p_4} = 5

\|\vec{p_4} - \vec{p_1}\| = \sqrt{(5 - (-3))^2} = 8,\ \|x\|\vec{d} + \|y\|\vec{d} = \|-1\| + \|4\| = 5

Can this be done with points in higher dimensions and where \vec{o} is not necessarily a zero vector? How would I go about calculating \|x\|\vec{d} + \|y\|\vec{d}? The euclidean distance part is easy.

Thanks.