How to solve an equation with vectors

Question

I have the equation $s = |v + at|$ where $v$ and $a$ are 2 dimensional vectors, $s$ and $t$ are scalars.

I am trying to rearrange this equation to solve for t, this should have two (or zero) solutions.

I can break out the vector components ($a$ = ($a$, $b$), $v$ = ($v$, $u$)) to get the equation $s = (v + at)^2 + (u + bt)^2$. However, when I solve this on wolfram-alpha I get two very large equations for the solution. I wonder if the solutions will be simpler when represented with vector operations?

If this is possible to solve, please provide the steps.

The vector operation most helpful in this situation is: $x\cdot x=|x|^2$. — Matthew Leingang, Mar 20 '18 at 15:34
$s^2=|v+at|^2=|v|^2+2t(v,a)+t^2|a|^2$. Or wait, from what you are saying it looks like $|\cdot|$ might not be the norm of the vector. Two equations? What is $|\cdot|$? The norm, or maybe componentwise absolute value? — SphericalTriangle, Mar 20 '18 at 15:34
@SphericalTriangle the magnitude, but even a solution for magnitude squared is fine — t123, Mar 20 '18 at 15:45
Then you get only one equation $s^2=|v|^2+2t(v,a)+t^2|a|^2$. — SphericalTriangle, Mar 20 '18 at 15:49

score 1 · Accepted Answer · answered Mar 20 '18 at 15:56

Hint: Note that

$$s^2=|v+at|^2=(v+at)\cdot(v+at)=|v|^2+2(v\cdot a)t+t^2|a|^2.$$

Given that $|a|\not=0$, we can rearrange to

$$0=t^2+\underbrace{\frac{2(v\cdot a)}{|a|^2}}_{=:\,p}t+\underbrace{\frac{|v|^2-s^2}{|a|^2}}_{=:\,q}=t^2+pt+q.$$

You should be able to solve a quadratic equation for its two solutions $t_{1/2}$.

How to solve an equation with vectors

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