Decompose v into two vectors v1 and v2 where v1 is parallel to w and v2 is orthogonal to w v = i + 5 j w = − 2 i + j

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Decompose v into two vectors v1 and v2 where v1 is parallel to w and v2 is orthogonal to w   v  =  i  +  5  j  w  =  −  2  i  +  j

gutsy8v · Accepted Answer

Step 1      V    1   can be obtained by the vector projection of v onto w by by using the formula,      V    1    =  P  r  o      j          w        v  =            V      .      W                      |                    |            W              |                              |                2              WStep 2Here we have,   v  =  i  +  5  j  ,  w  =  −  2  i  +  jSo,       V    1    =  P  r  o      j          w        v  =            (      i      +      5      j      )      ⋅      (      −      2      i      +      j      )              (              (        −        2                  )          2                +                  i          2                            )        2              wStep 2  ⇒      V          1        =            1      (      −      2      )      +      5      (      1      )              (              5                    )        2              w  =            −      2      +      5        5    w  ⇒      V          1        =      3    5    w  =      3    5    (  −  2  i  +  j  )  ⇒      V    1    =            −      6        5    i  +      3    5    jNow       V          2      , which is orthogonal to w can be found by subtracting vector       V          1       from vector v. Because,   v  =      v    1    +      v    2  Step 3  ⇒      V          2        =  V  −      V          1        ⇒      V          2        =  i  +  5  j  −  (            −      6        5    i  +      3    5    j  )  ⇒      V          2        =  i  +  5  j  +      6    5    i  −      3    5    j  =  (  i  +      6    5    i  )  +  (  5  j  −      3    5    j  )  =      11    5    i  +      22    5    jSo, the vector       V    1   which is parallel to w is             −      6        5    i  +      3    5    j and the vector       V    2   which is orthogonal to w is       11    5    i  +      22    5    j

Decompose v into two vectors v1 and v2 where v1 is parallel to w and v2 is orthogonal to w v = i + 5j w = -2i+j

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