# Parallel and orthogonal

I need math help in this problem: a=(-5, 5 3) b=(-2,-4,-5) (they are vectors)
Decompose the vector b into b=v+w where v is parallel to a and w is orthogonal to a, find v and w

Result

v1 =  2.119
v2 =  -2.119
v3 =  -1.271
w1 =  -4.119
w2 =  -1.881
w3 =  -3.729

#### Solution:

$a = (-5, 5, 3) \ \\ b = (-2,-4,-5) \ \\ \ \\ b = v+w \ \\ v \parallel a = > v = k a \ \\ \ \\ w \perp a = > w.a = 0 \ \\ \ \\ \ \\ -2 = v_{ 1 }+w_{ 1 } \ \\ -4 = v_{ 2 }+w_{ 2 } \ \\ -5 = v_{ 3 }+w_{ 3 } \ \\ v_{ 1 } = -5 \cdot \ k \ \\ = 2.119 \ \\ v_{ 2 } = 5 \cdot \ k \ \\ v_{ 3 } = 3 \cdot \ k \ \\ w_{ 1 } \cdot \ (-5)+w_{ 2 } \cdot \ 5 + w_{ 3 } \cdot \ 3 = 0 \ \\ \ \\ v_{ 1 }+w_{ 1 } = -2 \ \\ v_{ 2 }+w_{ 2 } = -4 \ \\ v_{ 3 }+w_{ 3 } = -5 \ \\ 5k+v_{ 1 } = 0 \ \\ 5k-v_{ 2 } = 0 \ \\ 3k-v_{ 3 } = 0 \ \\ 5w_{ 1 }-5w_{ 2 }-3w_{ 3 } = 0 \ \\ \ \\ k = \dfrac{ -25 }{ 59 } \doteq -0.423729 \ \\ v_{ 1 } = \dfrac{ 125 }{ 59 } \doteq 2.118644 \ \\ v_{ 2 } = \dfrac{ -125 }{ 59 } \doteq -2.118644 \ \\ v_{ 3 } = \dfrac{ -75 }{ 59 } \doteq -1.271186 \ \\ w_{ 1 } = \dfrac{ -243 }{ 59 } \doteq -4.118644 \ \\ w_{ 2 } = \dfrac{ -111 }{ 59 } \doteq -1.881356 \ \\ w_{ 3 } = \dfrac{ -220 }{ 59 } \doteq -3.728814 \ \\$
$v_{ 2 } = (-2.1186) = - \dfrac{ 125 }{ 59 } \doteq -2.1186 = -2.119$
$v_{ 3 } = (-1.2712) = - \dfrac{ 75 }{ 59 } \doteq -1.2712 = -1.271$
$w_{ 1 } = (-4.1186) = - \dfrac{ 243 }{ 59 } \doteq -4.1186 = -4.119$
$w_{ 2 } = (-1.8814) = - \dfrac{ 111 }{ 59 } \doteq -1.8814 = -1.881$
$w_{ 3 } = (-3.7288) = - \dfrac{ 220 }{ 59 } \doteq -3.7288 = -3.729$ Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

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