# Distance an arbitrary point is found along a given vector

Say I have a vector in 2D space defined by two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$: $$\vec{v}=(x_2 - x_1, y_2 - y_1)$$ I would like to find how far along that vector an arbitrary point $$(x_3, y_3)$$ is. This very woolly language$$^*$$, so I've attempted to create a diagram showing the sitation.

In this diagram, the quantity I'm interested in $$a$$, which I can calculate using Pythagoras' theorem if I know $$b$$ and $$c$$. I know $$c$$, which is the length of vector $$(x_3 - x_1,y_3 - y_1)$$, given by, $$c = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2}$$ So, now I need to calculate $$b$$: the length of a vector – that I'll call $$\vec{u}$$ – that is perpendicular to $$\vec{v}$$ and passes through point $$(x_3, y_3)$$. For $$\vec{u}$$ and $$\vec{v}$$ to be perpendicular the dot product must be zero. That is,

$$\vec{v}\cdot \vec{u}=0$$

$$(x_4-x_3)(x_2-x_1)+(y_4-y_3)(y_2-y_1)=0$$

This is where I begin to falter: one equation with two unknowns, $$y_4$$ & $$x_4$$. I expect there is some obvious constraint on $$\vec{u}$$ that I should be using to eliminate an unknown, but my sleep-deprived mind is offering no help. Can someone point out what I've missed?



$$^*$$I really want to use the word project to describe how my arbitrary point $$(x_3, y_3)$$ is placed along that vector. Is this the correct terminology?

You are right in calling this a projection. If $$(x_1,y_1)$$ is the origin, then you can project $${\bf u} = (x_3,y_3)$$ onto $$\bf v$$ thus:

$${\rm proj}_{\bf v}{\bf u} = \frac{\bf u \cdot v}{\bf v \cdot v}{\bf v}.$$

If $$(x_1,y_1)$$ is not the origin, then just shift the frame of reference to make it so.

Consider the vectors $$v_1 = (x_2-x_1,y_2-y_1)$$ and $$v_2 = (x_3-x_1,y_3-y_1)$$. You can compute the cosine of the angle between these two vectors as follows:

$$\cos \theta = \frac{}{||v_1||||v_2||}$$,

where $$$$ is the dot-product and $$||.||$$ is the norm. Once you have done this computation, it is easy to see that $$a = c \cos \theta$$ and $$b = c \sin \theta$$.

The quantity $$a$$ is the projection you speak of and it is related to the dot-product as described above.