Ask Uncle Colin: A Vector Line

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to and Uncle Colin will do what he can.

Dear Uncle Colin,

I've got three points: $A$, with a position vector of $(2\bi + 4\bj)$, $B$, with a position vector of $(6\bi + 8\bj)$ and $C$, with a position vector of $(k\bi + 25\bj)$, and they all lie on the same straight line.

I have to find $k$, and I don't know where to start!

-- Points In Collinear Kerfuffle

Hello, PICK!

A good place to start would be to think about what a straight line is, as far as vector geometry goes: you can think of it all of the points 'in the same direction' from a given point -- or as any multiple of a specific direction vector added to a reference point.

In this case, if you took your reference point as $A$ and your direction vector as $\vec{AB}$, an equation of your line would be $$\mathbf{r} = (2\bi + 4\bj) + \lambda(4\bi + 4\bj)$$.

Now, you know that $\mathbf{r} = (k\bi + 25\bj)$ lies on this line, so $(k\bi + 25\bj) = (2\bi + 4\bj) + \lambda(4\bi + 4\bj)$ for some values of $k$ and $\lambda$.

We can split this out into two equations: in $\bi$, $k = 2 + 4\lambda$; in $\bj$, $25 = 4 + 4\lambda$.

The second equation gives $\lambda = \frac{21}{4}$, so $k = 23$.

This is the vector way of looking at it, PICK. There is a possibly simpler way, which is to look at the position vectors of $A$, $B$ and $C$ as points in the Cartesian plane.

What's the equation of the line through (2,4) and (6,8)? It's $y = x + 2$. So when $y=25$, like it does at point $C$, then $x = 23$, which is your value of $k$.

Hope that helps!

-- Uncle Colin

* Edited 2016-09-28 to fix broken LaTeX. Thanks to @christianp and @dragondodo for pointing it out.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


One comment on “Ask Uncle Colin: A Vector Line

  • Keri

    Hi Uncle Colin,
    Trying to solve a problem for my maths, can’t figure out what to substitute to get it into a quadratic form.
    5^x = 6 – 5^(1-x)

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