Find two vectors satisfying given conditions

Let

$A = ( 1,2,3,4,5), \qquad B = \left( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5} \right).$

Find two vectors $C, D \in \mathbb{R}^5$ such that

$B = C + 2D, \qquad D \cdot A = 0$

and $C$ is parallel to $A$ .

Let $C = (c_1, c_2, c_3, c_4, c_5)$ and $D = (d_1, d_2, d_3, d_4, d_5)$ . Then we have

$\begin{align*} B = C + 2D && \implies && c_1 + 2d_1 &= 1 \\ &&&& c_2 + 2d_2 &= 2 \\ &&&& c_3 + 2d_3 &= 3 \\ &&&& c_4 + 2d_4 &= 4 \\ &&&& c_5 + 2d_5 &= 5. \end{align*}$

We also have

$D \cdot A = 0 \quad \implies \quad d_1 + 2d_2 + 3d_3 + 4d_4 + 5d_5 = 0.$

And, since $C$ is parallel to $A$ there is a nonzero $x$ such that

$C = xA \quad \implies \quad c_1 = x, c_2 = 2x, c_3= 3x, c_4 = 4x, c_5 = 5x.$

So, putting these together we get expressions for the $d_i$ ,

$\begin{align*} d_1 &= \frac{1}{2}(1-x) \\ d_2 &= \frac{1}{2}\left( \frac{1}{2} - 2x \right) \\ d_3 &= \frac{1}{2}\left( \frac{1}{3} - 3x \right) \\ d_4 &= \frac{1}{2}\left( \frac{1}{4} - 4x \right) \\ d_5 &= \frac{1}{2}\left( \frac{1}{5} - 5x \right). \end{align*}$

Therefore,

$\begin{align*} &&\frac{1}{2}(1-x) + \left( \frac{1}{2} - 2x \right) + \frac{3}{2} \left( \frac{1}{3} - 3x \right) + 2 \left( \frac{1}{4} - 4x \right) + \frac{5}{2} \left( \frac{1}{5} - 5x \right) &= 0 \\ \implies && 1-x + 2 \left( \frac{1}{2} - 2x \right) + 3 \left( \frac{1}{3} - 3x \right) + 4 \left( \frac{1}{4} - 4x \right) + 5\left( \frac{1}{5} - 5x \right) &= 0 \\ \implies && 55x &= 5 \\ \implies && x &= \frac{1}{11}. \end{align*}$

Therefore, we have

$C = \frac{1}{11} (1,2,3,4,5), \qquad D = \left( \frac{5}{11}, \frac{7}{44}, \frac{1}{33}, \frac{-5}{88}, \frac{-7}{55} \right).$

Stumbling Robot

Find two vectors satisfying given conditions

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