Given two vectors a → = ( 5 , 7 , − 3 ) and b → = ( 1 , 0 , − 3 )Find the projection of a → onto b → So we first find | a → | = 83 and then | b → | = 10 We find cos ⁡ θ using a → ⋅ b → = | a → | | b → | c o s θWhich is: 14 = 83 ∗ 10 cos ⁡ θ ⟺ cos ⁡ θ = 14 830 Now we want to find a → b ^ where b ^ = b → | b → | so we get: a → ⋅ b → | b → | = 14 10 but it should have a direction, so the answer is 14 10 b ^ ? but we have used b ^ = b → | b → | in the calculation

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Given two vectors             a      →        =  (  5  ,  7  ,  −  3  ) and             b      →        =  (  1  ,  0  ,  −  3  )Find the projection of             a      →        onto             b      →      So we first find       |              a      →            |    =      83   and then       |              b      →            |    =      10  We find   cos  ⁡  θ using            a      →        ⋅            b      →        =      |              a      →            |        |              b      →            |    c  o  s  θWhich is:  14  =      83    ∗      10    cos  ⁡  θ    ⟺    cos  ⁡  θ  =      14          830      Now we want to find             a      →                  b      ^       where             b      ^        =                    b        →                            |                              b          →                            |             so we get:                              a          →                    ⋅                        b          →                                    |                              b          →                            |              =      14          10      but it should have a direction, so the answer is       14          10                  b      ^      ? but we have used             b      ^        =                    b        →                            |                              b          →                            |             in the calculation

Yaretzi Melendez · Accepted Answer

Yes, the projection of             a      →        onto             b      →       is equal to  (            a      →        ⋅            b      ^        )  ⋅            b      ^      where             b      ^       is the normalized version of vector             b      →       . Why do you consider this wrong? Sure, you use             b      ^       both to calculate the length and the direction of the projection, but why exactly do you consider that wrong?If you want a different formula, the projection of             a      →        onto             b      →       can also be written as                              a          →                    ⋅                        b          →                                    |                              b          →                                      |                2              ⋅            b      →      or as                              a          →                    ⋅                        b          →                                              b          →                    ⋅                        b          →                      ⋅            b      →      since all three expressions return the same vector.

Given two vectors vec(a) =(5,7,−3) and vec(b) =(1,0,−3) Find the projection of vec(a) onto vec(b)

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