Question

The vectors are u= 6i -10j+4k and v= 14i +2j-4k.find the projection of u and v and the vector component of u orthogonal v.​

Answer 1

Answer:

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Answer 2

Answer:

The projection of$\overrightarrow{u}$on $\overrightarrow{v}$ is $<\dfrac{28}{9},\dfrac{4}{9},\dfrac{-8}{9} >$ and the answer for the vector component is option (c), that is $\frac{26}{9}i-\frac{94}{9}j+\frac{44}{9}k$

Step-by-step explanation:

The question is to find the projection of $\overrightarrow{u}$ and $\overrightarrow{v}$, and the vector component of

Firstly write down the components, $\overrightarrow{u}=< 6,-10,4>$

                                              $\overrightarrow{v}= <14,2,-4 >$

The formula to find the projection of$\overrightarrow{u}$on $\overrightarrow{v}$ = $\dfrac{\overrightarrow{u}\cdot \overrightarrow{v}}{\left| \overrightarrow{v}\right| ^{2}}\left \overrightarrow{v}\right$

                                            $=\dfrac{\left( 6\times 14\right) +\left( -10\times 2\right) +\left( 4\times -4\right) }{14^{2}+2^{2}+\left( -4^{2}\right) }\overrightarrow{v}$                                                       $=\dfrac{48}{216}< 14,2,-4>$

                                             $=\dfrac{2}{9} <14,2,-4 >$

                                             $= <\dfrac{28}{9},\dfrac{4}{9},\dfrac{-8}{9} >$

So this is the answer to projection. Now we have to find the vector component of $\overrightarrow{u}$ orthogonal $\overrightarrow{v}$. For that, we have to subtract the projection of

Vector component of $\overrightarrow{u}$ orthogonal $\overrightarrow{v}$, $\overrightarrow{w}$$=\overrightarrow{u}$-projection of

                                                     

                   $\overrightarrow{w}= <6,-10,4>-<\frac{28}{9},\frac{4}{9},\frac{-8}{9}>$

  To make the subtraction easy, multiply u components with 9. Then,

         $\overrightarrow{w}= <\frac{54}{9} ,\frac{-90}{9} ,\frac{36}{9} >-<\frac{28}{9},\frac{4}{9},\frac{-8}{9}>$

            $= <\frac{26}{9},\frac{-94}{9},\frac{44}{9}>$

            $=\frac{26}{9}i-\frac{94}{9}j+\frac{44}{9}k$

This is the answer for the vector component. From the given options, Option c is the answer.