Multi-Variable Calculus: Linear Combination of Vectors

  1. Dembadon

    Dembadon 662
    Gold Member

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

    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. :frown:
     
  2. jcsd
  3. Dick

    Dick 25,823
    Science Advisor
    Homework Helper

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

    Dembadon 662
    Gold Member

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