# 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:

$a = (-5, 5, 3) \ \\ b = (-2,-4,-5) \ \\ \ \\ b = v+w \ \\ v \parallel a = > v = k a \ \\ \ \\ w \perp a = > w.a = 0 \ \\ \ \\ \ \\ -2 = v_{ 1 }+w_{ 1 } \ \\ -4 = v_{ 2 }+w_{ 2 } \ \\ -5 = v_{ 3 }+w_{ 3 } \ \\ v_{ 1 } = -5 \cdot \ k \ \\ = 2.119 \ \\ v_{ 2 } = 5 \cdot \ k \ \\ v_{ 3 } = 3 \cdot \ k \ \\ w_{ 1 } \cdot \ (-5)+w_{ 2 } \cdot \ 5 + w_{ 3 } \cdot \ 3 = 0 \ \\ \ \\ v_{ 1 }+w_{ 1 } = -2 \ \\ v_{ 2 }+w_{ 2 } = -4 \ \\ v_{ 3 }+w_{ 3 } = -5 \ \\ 5k+v_{ 1 } = 0 \ \\ 5k-v_{ 2 } = 0 \ \\ 3k-v_{ 3 } = 0 \ \\ 5w_{ 1 }-5w_{ 2 }-3w_{ 3 } = 0 \ \\ \ \\ k = \dfrac{ -25 }{ 59 } \doteq -0.423729 \ \\ v_{ 1 } = \dfrac{ 125 }{ 59 } \doteq 2.118644 \ \\ v_{ 2 } = \dfrac{ -125 }{ 59 } \doteq -2.118644 \ \\ v_{ 3 } = \dfrac{ -75 }{ 59 } \doteq -1.271186 \ \\ w_{ 1 } = \dfrac{ -243 }{ 59 } \doteq -4.118644 \ \\ w_{ 2 } = \dfrac{ -111 }{ 59 } \doteq -1.881356 \ \\ w_{ 3 } = \dfrac{ -220 }{ 59 } \doteq -3.728814 \ \\$
$v_{ 2 } = (-2.1186) = - \dfrac{ 125 }{ 59 } \doteq -2.1186 = -2.119$
$v_{ 3 } = (-1.2712) = - \dfrac{ 75 }{ 59 } \doteq -1.2712 = -1.271$
$w_{ 1 } = (-4.1186) = - \dfrac{ 243 }{ 59 } \doteq -4.1186 = -4.119$
$w_{ 2 } = (-1.8814) = - \dfrac{ 111 }{ 59 } \doteq -1.8814 = -1.881$
$w_{ 3 } = (-3.7288) = - \dfrac{ 220 }{ 59 } \doteq -3.7288 = -3.729$

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