Sum of a vector parallel to b and a vector orthogonal to b

Vectors are

$\displaystyle

\begin{array}{l}

\underline v = < 2, - 4 > \\

\underline b = < 1,1 > \\

\end{array}

$

As far as a parallel vector to vector b went it would be just in the same direction and thus for expressing vector v as the sum of a vector parallel to vector b. I thought it would be

$\displaystyle

\underline v + \underline b = \left\langle {(2 + 1)} \right.,\left. {( - 4 + 1)} \right\rangle = \left\langle 3 \right.,\left. { - 3} \right\rangle

$

and for expressing the vector v as the sum of a vector orthogonal to vector b would be just

$\displaystyle

\bot \underline {b = \left\langle { - 1,\left. 1 \right\rangle } \right.}

$

and so

$\displaystyle

\underline v + \bot \underline b = \left\langle {(2 + ( - 1)),\left. {( - 4 + 1)} \right\rangle = \left\langle {1,\left. { - 3} \right\rangle } \right.} \right.

$

However the answer given for the orthogonal part of the question is $\displaystyle

\left\langle { - 1,\left. { - 1} \right\rangle } \right.

$

I'm not sure what I have done wrong in the later part of the question.