# Math Help - A Question with vectors

1. ## A Question with vectors

Sorry, I didn't know where to post this. But here is the question.

A surveyor measures the location of point and determines that . He wants to determine the location of a point so that and . Assuming that the point lies to the east of the point , what are the (a) - and (b) - coordinates of point ?

Any help would be great!

2. Here's one way to do it. Suppose the location of B is given by $\mathbf{r}_{\textsf{\tiny OB}} = p\mathbf{i} + q\mathbf{j}$. Then Pythagoras' theorem tells you that $p^2+q^2=2400^2$ and $(p-297)^2 + (q-743)^2 = 1647^2$.

Multiply out the brackets in the second of those equations, subtract the first equation, and you'll get a simple linear relation between p and q. Solve this for q, substitute that into the equation $p^2+q^2=2400^2$, and you'll have a quadratic equation for p. That will have two solutions, and you want the larger one (because B has to lie to the east of A).

3. I can't seem to figure it out, it looks easy. But the numbers are getting to big and stuff. Is there an easier way to do it?

4. bump, anyone please?

5. I agree, the numbers are not nice.

We have the equations

$p^2+q^2=2400^2$,
$p^2-594p+297^2 + q^2-1486q+743^2 = 1647^2$.

Subtract the first of these from the second, getting

$594p + 1486q = 2400^4+743^2+297^2-1647^2 = 3687649$. (*)

Now multiply both sides of the first equation by $1486^2$:

$1486^2p^2 + (3687649-594p)^2 = 2400^2\times1486^2$,

which simplifies(?) to $2561032p^2 -4.380927\times10^9p + 8.7954619\times10^{11} = 0$.

Plug those coefficient numbers into the formula for solving a quadratic, and you'll find that the larger root is (approximately)

$p = \frac{4.380927\times10^9 + 3.1909775\times10^9}{5122064} \approx 1478$.

Having found p, substitute its value into (*) to get $q\approx1891$.

On second thoughts, if I came across that problem in real life, I would solve it on a scale diagram, drawing a circle with a radius representing 2400m, centred at O, and one with radius repesenting 1647m, centred at A, and measuring the coordinates of the point(s) where the circles intersect. That would be quicker than doing the calculation, and would probably give you a sufficiently accurate answer for practical purposes.