Find |4a - 5b|, |a| = 1, |b|=2, a*b = -(1/3)*|a| |b|

[soopo]

Homework Statement

Find |4a - 5b|, |a| = 1, |b|=2, a*b = -(1/3)*|a| |b|
where

a and b are vectors.

The Attempt at a Solution

I use big letters (X,Y) for the vector `a`, while small letters
for the vertor `b` (x,y).

(data) =>
(1) X^2 + Y^2 = 1
(2) x^2 + y^2 = 4
(3) Xx + Yy = -2/3

--- Data processing

(11) X = (1 - Y)^.5
(22) x = (1 - y)^.5
(33) (Yy)^.5 + Yy - y^2 - 4y^2 = 0

Solving (33) gives for Y
(44) $$Y = -.5 \pm .5 * \sqrt ( \frac { 17y^2 -1} {y^2 -1} )$$

I am not certain whether "brute force" is the best tool here.

 

[lanedance]

i don't think you have to explictly solve for (x,y) & (X,Y) in fact i doubt you can...

you eant to find the length of a linear combination of the vectors, which will eb independent fo any ratation to the axis you make, so why not make a = (1,0) then use the dot product to give the angle between a & b to fi9nd b

 

[HallsofIvy]

$|4a-5b|^2$$= (4a-5b)\cdot(4a- 5b)$$= 16|a|^2- 40a\cdot b+ 25|b|^2$

That's pretty much all you need, isn't it?