Determine the constant α ∈ R and the vector w ∈ R 3 that is perpendicular to the vector u such that v = α u + w

Name: Determine the constant α ∈ R and the vector w ∈ R 3 that is perpendicular to the vector u such that v = α u + w
Uploaded: 2022-02-23T17:22:57+00:00
Description: Determine the constant α ∈ R and the vector w ∈ R 3 that is perpendicular to the vector u such that v = α u + w

Question

Determine the constant   α  ∈      R   and the vector   w  ∈            R        3   that is perpendicular to the vector u such that   v  =  α  u  +  w

boursecasa2je · Accepted Answer

You have   v  =  α  u  +  w. Now, dot product by u gives  v  ⋅  u  =  α  u  ⋅  u  +  w  ⋅  uWe know   w  ⋅  u  =  0 so the equation gives  3  =  α  9so  α  =            1      3      Knowing α it is straightforward that  w  =      (                  7        3              ,                  5        3              ,    −                  4        3              )

Anton Huynh · Accepted Answer

Let   w  =  (      w    1    ,      w    2    ,      w    3    ). If w is perpendicular to u, the equation   w  ⋅  u  =  0 must be satisfied, that is,   2      w    1    −  2      w    2    +      w    3    =  0. As you see there are lots of solutions.If you want   v  =  α  u  +  w, then you have another equation, in this cases a system of equations indeed, namely  3  =  2  α  +      w    1    1  =  2  α  +      w    2    −  1  =  α  +      w    3  Adding the equation   2      w    1    −  2      w    2    +      w    3    =  0 from above you&#039;ve got a linear system of 4 equations and 4 unknowns, so there is precisely one solution that I&#039;m sure you&#039;re able to find.

Determine the constant alpha in RR and the vector w in RR^3 that is perpendicular to the vector u such that v = alpha u + w.

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