# Multi-Variable Calculus: Linear Combination of Vectors

1. Aug 31, 2011

### Dembadon

I would like to check my work with you all.

1. The problem statement, all variables and given/known data

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$\vec{w}$.

2. Relevant equations

Standard Unit Vectors:

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

3. 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.

2. Aug 31, 2011

### Dick

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

3. Aug 31, 2011

### Dembadon

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

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