Solving for Largest X in Three Vectors with Magnitude Resulting in 17

[MMCS]

Three vectors

a = xi - 3j + 9K
b = -3i + xj - 10K
c = 9i - 10j + xk

Find the largest value of x for which the magnitude of the resultant is equal to 17.
I am given the correct answer of 8.564 but i don't know how to get that.

I have no working out to show for this as i don't know where to start, but some advice would be helpful

Thanks

 

[haruspex]

How do you work out the resultant of two vectors?

 

[HallsofIvy]

I'm not sure that mentioning the "resultant of two vectors" will help here at all- except, of course, to point out that it is just as easy to find the resultant of three vectors!

MMCS, do you understand that the "resultant" of any number of vectors is just their sum? Do you understand that you can sum vectors "component wise"- that is just by adding the i, j, and k components separately? What do you get for the resultant of these three vectors? (There will, of course, be an "x" in each component.)

What is the magnitude of that resultant? That will be a forumula with terms involving "x". Set that equal to 17 and solve for x.

 

[MMCS]

ok so once i complete the addition of i, j and k and then square the brackets and squared 17 to eliminate square root i get

(x^2 + 12x + 36) + ( X^2 + 14x + 49) + (x^2 - 2x + 1) = 289

Addition of brackets

x^6 + 24x + 86 = 289

Is this correct? can this be solved?

 

[scurty]

ok so once i complete the addition of i, j and k and then square the brackets and squared 17 to eliminate square root i get

(x^2 + 12x + 36) + ( X^2 + 14x + 49) + (x^2 - 2x + 1) = 289

Addition of brackets

x^6 + 24x + 86 = 289

Is this correct? can this be solved?

You summed the j hat components wrong. Also, $x^2 + x^2 + x^2 = 3 x^2$

 

[haruspex]

ok so once i complete the addition of i, j and k and then square the brackets and squared 17 to eliminate square root i get

(x^2 + 12x + 36) + ( X^2 + 14x + 49) + (x^2 - 2x + 1) = 289

Addition of brackets

x^6 + 24x + 86 = 289

Is this correct? can this be solved?

You shouldn't have an x^6 there. try that last step again.

 

[MMCS]

You summed the j hat components wrong. Also, $x^2 + x^2 + x^2 = 3 x^2$

Oh right i never noticed that, good spot, so now i have,

3x^2 - 16x + 206

 

[scurty]

Yes. Assuming you didn't subtract the 289 yet.

 

[MMCS]

Thats it, finally got it, Thanks for you help!