Multi-Variable Calculus: Linear Combination of Vectors

Dembadon

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,a-b>[/itex].

Find Scalars:

[itex]a+b = 2[/itex].
[itex]a-b = 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 lining-up very well.

Dick

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

Dembadon

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