Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

Consider the following. u = (−8i − 4j − 2k), v = (4j + 4k ) ?

a) Find the projection of u onto v.

b) Find the vector component of u orthogonal to v.

I keep getting the wrong answers. The last one I got was 324/90 i as part one of part a (finding the projection) but it wont let me move on to the next parts until I get this right. Can someone please help?

1 Answer

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  • Alan
    Lv 7
    1 month ago

     projection of u onto v =  (u dot v) / |v|  * v      

    u dot v =    -8*0  +  -4 *4  +  -2*4   =  -16  - 8= -24  

    |v|  =   sqrt(  16 + 16)  = sqrt(32)     =  sqrt(16)sqrt(2) = 4*sqrt(2) 

    |v|  =   4*sqrt(2)  

    (u dot v ) / |v|  =   -24/ (4*sqrt(2) =  -6/sqrt(2) = -6*sqrt(2)/2  = -3*sqrt(2) 

    proj. u on v =  (-3*sqrt(2))*(4j  + 4k)   =  -12*sqrt(2) j   - 12*sqrt(2)k 

    proj. u on v  =  -12*sqrt(2) j - 12*sqrt(2)k 

    (b) orthogonal component  

    u orthogonal =   u  -   proj u on v  =  

    = -8i   -4j  - 2k   -   (-12*sqrt(2) )j   - (-12*sqrt(2))k

    u orthogonal to v   

    =  -8i   + (12*sqrt(2) - 4)j    + (12*sqrt(2) - 2)k   

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