Multi-Variable Calculus: Linear Combination of Vectors

Dembadon
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I would like to check my work with you all. :smile:

Homework Statement



Let
\vec{u} = 2\vec{i}+\vec{j},
\vec{v} = \vec{i}+\vec{j}, and
\vec{w} = \vec{i}-\vec{j}.

Find scalars a and b such that \vec{u} = a\vec{v}+ b\vec{w}.

Homework Equations



Standard Unit Vectors:

\vec{i} = <1,0>.
\vec{j} = <0,1>.

The Attempt at a Solution



Compute vectors:

\vec{u} = 2<1,0>+<0,1>=<2,1>.
\vec{v} = <1,0>+<0,1>=<1,1>.
\vec{w} = <1,0>-<0,1>=<1,-1>.

Setup Scalars:

<2,1> = a<1,1>+b<1,-1>.
<2,1> = <a,a>+<b,-b>.
<2,1> = <a+b,a-b>.

Find Scalars:

a+b = 2.
a-b = 1.

Thus, a = 3/2 and b = 1/2.

Final answer:

\vec{u} = \frac{3}{2}\vec{v}+\frac{1}{2}\vec{w}.

Note: Sorry my vector arrows aren't lining-up very well. :frown:
 
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Certainly correct that (3/2)v+(1/2)*w=u. No question, you are just checking?
 
Dick said:
Certainly correct that (3/2)v+(1/2)*w=u. No question, you are just checking?

Yes, just checking my work. Thank you for verifying. :smile:
 
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