Question

Let v = 5i + 2j. find a vector, w, that is half the length of v and points in the opposite direction of v.
A = -10/sqrt(29)i - 4/sqrt(29)j
b = 5/(2sqrt(29))i + 1/sqrt(29)j
c. = 10/sqrt(29)i + 4/sqrt(29)j
d = -5 / 2sqrt(29)i - 1/sqrt(29)j

Answer 1

What is the magnitude of v?

Answer 2

(length of vector v)
use pythagorean theorem

Answer 3

so take the same vector (5,2) and reverse it
(5,20->(-5,-2)
now,
norm = root(25+4)=root29
half = root29/2
now, we divide by the old norm and multiply by the new
(-5,-2) *root29 *2/root29
gives
0.5 * (-5,-2)

Answer 4

none of your choices make sense.
to get the opposite direction multiply the vector by -1
to get ½ the magnitude, multiply by ½
i.e. -½ * < 5,2> = <-2.5,-1>

Answer 5

Woops phi :O
Cutting the components in half doesn't cut the length of v directly in half.
I guess it would actually cut the length of v into 1/4 of it's original length,
since the legs are squared.... or something... like.. that.

Answer 6

You figure this one out ok studygeek? :x

Answer 7

@zepdrix
see http://www.wolframalpha.com/input/?i=++norm%28%3C-2.5%2C-1%3E%29%2Fnorm%28%3C5%2C2%3E%29

or compute the length of <-2.5,-1> and <5,2> and find the ratio of their lengths

Answer 8

or more intuitively, if you have a unit vector pointing in any direction, if you multiply it by n you get a vector n units long. that idea works for non-unit vectors... you scale its length by n

Answer 9

I think i got it. Thanks for the help.

Answer 10

Mmm yah that seems to make sense phi :(
hmm uh oh my brain just esploded

Answer 11

which still leaves the problem that none of the choices seem appropriate.

Answer 12

See how they divided by sqrt29 in each option?
It almost seems like what they did is found the unit vector in the direction of v,
then the options are in terms of length 2 and 1 i guess..?
like c)\[\Large\rm 2\left<\frac{5}{\sqrt{29}},~\frac{2}{\sqrt{29}}\right>\]
That's a vector in the direction of v, with length 2, right?
Certainly not half of v though...

Answer 13

yes. If they meant, with length of 2 (or length of ½) then we would expect to scale the original to unit length, then scale by the new length

Answer 14

hmm weird :D

Answer 15

I would point this one out to the teacher

Answer 16

find theta
then find -theta
then find the actual angle created there by doing -theta+180