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MultiVariable Calculus: Linear Combination of Vectors 
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#1
Aug3111, 10:30 PM

PF Gold
P: 641

I would like to check my work with you all.
1. The problem statement, all variables and given/known data Let [itex]\vec{u} = 2\vec{i}+\vec{j}[/itex], [itex]\vec{v} = \vec{i}+\vec{j}[/itex], and [itex]\vec{w} = \vec{i}\vec{j}[/itex]. Find scalars a and b such that [itex]\vec{u} =[/itex] a[itex]\vec{v}+[/itex] b[itex]\vec{w}[/itex]. 2. Relevant equations Standard Unit Vectors: [itex]\vec{i} = <1,0>[/itex]. [itex]\vec{j} = <0,1>[/itex]. 3. The attempt at a solution Compute vectors: [itex]\vec{u} = 2<1,0>+<0,1>=<2,1>[/itex]. [itex]\vec{v} = <1,0>+<0,1>=<1,1>[/itex]. [itex]\vec{w} = <1,0><0,1>=<1,1>[/itex]. Setup Scalars: [itex]<2,1> = a<1,1>+b<1,1>[/itex]. [itex]<2,1> = <a,a>+<b,b>[/itex]. [itex]<2,1> = <a+b,ab>[/itex]. Find Scalars: [itex]a+b = 2[/itex]. [itex]ab = 1[/itex]. Thus, a = 3/2 and b = 1/2. Final answer: [itex]\vec{u} = \frac{3}{2}\vec{v}+\frac{1}{2}\vec{w}[/itex]. Note: Sorry my vector arrows aren't liningup very well. 


#2
Aug3111, 10:35 PM

Sci Advisor
HW Helper
Thanks
P: 25,251

Certainly correct that (3/2)v+(1/2)*w=u. No question, you are just checking?



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