Vectors

v =  < 1, - 1 >

w = < 1, 2  >

v1   = [ v (dot) w  ] / l w l^2  *  < w >

v(dot) w   =   1* 1 + 2* -1   =  1 - 2    =   -1

l w l  =  sqrt (1^2 + 5^2)   =  sqrt (5)           l w l ^2   = 5

So......

v1   =   -1 /  (5)  *  <1,  2 >     =   < -1/5, -2/ 5  >

v2  =  v - v1  =  < 1 - -1/5, -1 - - 2/5 >  =  < 1 + 1/5, -1 + 2/5 > =   < 6/5,-3/5 >

Check

Sum of v1 and v2    =  < -1/5 + 6/5, -2/5 + - 3/5 >   =  < 1  -1 >    = v

Decompose v into two vectors v1 and v2,

where v1 is parallel to w and v2 is orthogonal to w.

v = i - j, w = i + 2j

\(\vec{v} = \binom{1}{-1}\\ \vec{w} = \binom{1}{2}\)

\(\begin{array}{rcll} \vec{v_1} &=& \lambda \cdot \vec{w} \\ \vec{v_2} &=& \mu \cdot \vec{w_\perp} \\ \hline \vec{v}=\vec{v_1}+\vec{v_2} &=& \lambda \cdot \vec{w} + \mu \cdot \vec{w_\perp} \\ \vec{v} &=& \lambda \cdot \vec{w} + \mu \cdot \vec{w_\perp} \quad &| \quad \cdot \vec{w} \\ \vec{v}\cdot \vec{w} &=& \lambda \cdot \vec{w}\cdot \vec{w} + \mu \cdot \vec{w_\perp} \cdot \vec{w} \quad &| \quad \vec{w_\perp} \cdot \vec{w} = 0 \quad \vec{w}\cdot \vec{w} = w^2 = 1^2+2^2 = 5\\ \vec{v}\cdot \vec{w} &=& \lambda \cdot w^2 +0 \\ \vec{v}\cdot \vec{w} &=& \lambda \cdot w^2 \quad &| \quad : w^2 \\ \lambda &=& \frac{ \vec{v}\cdot \vec{w} } {w^2} \\ \mathbf{ \vec{v_1} } &\mathbf{=}&\mathbf { \left( \frac{ \vec{v}\cdot \vec{w} } {w^2} \right) \cdot \vec{w} } \\ \mathbf{ \vec{v_2} } &\mathbf{=}&\mathbf { \vec{v} - \vec{v1} }\\ \hline \end{array} \)

\(\begin{array}{|rcll|} \hline \mathbf{ \vec{v_1} } &\mathbf{=}&\mathbf { \left( \frac{ \vec{v}\cdot \vec{w} } {w^2} \right) \cdot \vec{w} } \\ & = & \left( \frac{ \binom{1}{-1}\cdot \binom{1}{2} } {5} \right) \cdot \binom{1}{2} \\ & = & \left( \frac{ 1-2 } {5} \right) \cdot \binom{1}{2} \\ & = & \left( \frac{ -1 } {5} \right) \cdot \binom{1}{2} \\ \mathbf{ \vec{v_1} } &\mathbf{=}&\mathbf { \binom{-0.2}{-0.4} } \\\\ \mathbf{ \vec{v_2} } &\mathbf{=}&\mathbf { \binom{1}{-1} - \vec{v1} }\\ & = & \binom{1}{-1} - \binom{-0.2}{-0.4} \\ & = & \binom{1}{-1} + \binom{0.2}{0.4} \\ & = & \binom{1+0.2}{-1+0.4} \\ \mathbf{ \vec{v_2} } &\mathbf{=}&\mathbf { \binom{1.2}{-0.6} } \\ \hline \end{array} \)