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Find vectors satisfying given conditions

by RoRi

Let

    \[ A = (2,-1,2), \qquad B = (1,2,-2) \]

be two vectors in \mathbb{R}^3. Find vectors C,D \in \mathbb{R}^3 such that

    \[ A = C+D, \qquad B \cdot D = 0 \]

and C is parallel to B.

Let C = (c_1, c_2, c_3) and D = (d_1, d_2, d_3). Then

    \[ A = C + D \quad \implies \quad c_1 + d_1 = 2, \quad c_2 + d_2= -1, \quad c_3 + d_3 = 2. \]

Next,

    \[ B \cdot D = 0 \quad \implies \quad d_1 + 2d_2 - 2d_3 = 0. \]

Finally, C parallel to B implies that there exists some nonzero x such that

    \[ C = xB \quad \implies \quad c_1 = x, c_2 = 2x, c_3 = -2x. \]

Putting these together we have

    \begin{align*} c_1 + d_1 &= 2 &&& x + d_1 &= 2 &&& d_1 &= 2 - x\\ c_2 + d_2 &= -1 & \implies && 2x + d_2 &= -1 & \implies && d_2 &= -1-2x\\ c_3 + d_3 &= 2 &&& -2x + d_3 &= 2 &&& d_3 &= 2 + 2x. \end{align*}

Therefore, we have

    \[ (2-x) + 2(-1-2x) - 2(2+2x) = 0 \quad \implies \quad x = -\frac{4}{9}. \]

Thus,

    \begin{align*} C &= \left( -\frac{4}{9}, -\frac{8}{9}, \frac{8}{9} \right) & D &= \left( \frac{22}{9}, -\frac{1}{9}, \frac{10}{9} \right) \\ C &= \frac{4}{9} (-1,-2,2), & D &= \frac{1}{9} (22,-1,10). \end{align*}

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