Dembadon
Gold Member
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I would like to check my work with you all. 
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}.
Standard Unit Vectors:
\vec{i} = <1,0>.
\vec{j} = <0,1>.
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.

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.
