vectors! (1 Viewer)

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hmm..anyone here done them before?

cos this is rather weird

Let u = (1,2), What is the largest possible area of a parallelogram spanned by u and w with w a unit vector? find a w that gives this max area

umm..max area?

we can't differentiate this, so what do i do? draw heaps of diagram?

 

[jm1234567890]

u x w gives the area

EDIT: that is the cross-product of u and v

 

[clive]

I havent done vectors for a while, so im sorry if this is incorrect

Area of parallelogram =| U x W | = |U||W|sin(theta)

where theta is the angle between vectors

W has to be a unit vector => |W| = 1
U = (1,2) => |U| = SQRT(5)

so, A = |SQRT(5)sin(theta)|
dA/dtheta = SQRT(5)cos(theta)
which is zero when theta = pi/2 or 3pi/2
do all the second derivative test etc to prove that is a max (not sure if it necessary though)

So, the angle that maximises area is pi/2 or 3pi/2

max possible area = SQRT(5) units

W needs to be perpendicular to U, so W.U = 0
(1,2).(a,b) = 0 => a = 2 b = -1, or a = -2 b = 1

W needs to be a unit vector though, so
W = ( 2/SQRT(5) , -1/(SQRT(5) )

or

W = ( - 2/SQRT(5) , 1/(SQRT(5) )