Given U is a vector with an initial point of (6,-4) and a terminal point of (-2, 8). How do you write u as a linear combination of the standard unit vector i and j?

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Answer 1 · 2017-02-04T13:43:45

$vecU=-8veci+12vecj$

$"label "(6,-4)" point"A; "and "(-2,8)" point" B$

so the position vectors for each point is as follows ( writing them as column vectors)

$vec(OA)=((6),(-4))$

$vec(OB)=((-2),(8))$

$vecU" has initial point "A " and terminal point "B$

$"in vector notation "vecU=vec(AB)$

so:

$vecU=vec(AB)=vec(AO)+vec(OB$

$vecU=vec(AB)=-vec(OA)+vec(OB$

$vecU=vec(AB)=-((6),(-4))+((-2),(8))$

$vecU=vec(AB)=((-8),(12))$

in $veci$ $"& "vecj "terms"$

$vecU=-8veci+12vecj$

Answer 2 · 2017-02-04T13:56:18

$-8veci+12vecj$

If $vecu=xveci+yvecj$ , the conventional notation is

$vecu = < x, y>$ , in Cartesian form, and

$= r< costheta, sintheta>$ , in polar form.

Here, $vecu=vec(AB)$ , where the position vectors

$vec(OA)=<6, -4> and vec(OB)=<-2, 8>$ , giving

$vecu=vec(AB)=vec(OB)-vec(OA)$

$=<-2,8> - <6, -4> = <((-2-6), (8-(-4))> = <-8, 12>$

$=-8veci+12vecj$