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#