4 vectors a, b, c and d.
a= 3i + 2j -6k
d= 2i - j + k
a is the sum of vectors b and c.
Find b and c so that b is PARALLEL to d and c is PERPENDICULAR to d.
Let vectors b and c be
$\displaystyle b = xi + yj + zk$
and
$\displaystyle c = pi + qj + rk$
Then
$\displaystyle a = b + c$
$\displaystyle 3i + 2j - 6k = (x + p)i + (y + q)j + (z + r)k$
Thus
$\displaystyle \begin{matrix} x + p = 3 \\ y + q = 2 \\ z + r = -6 \end{matrix}$
Now, b is parallel to d, so
$\displaystyle d = eb$
Thus
$\displaystyle 2i - j + k = e(xi + yj + zk)$
Thus
$\displaystyle \begin{matrix}ex = 2 \\ ey = -1 \\ e = z \end{matrix}$
And finally c is perpendicular to d, so
$\displaystyle c \cdot d = 0$
$\displaystyle (pi + qj + rk) \cdot (2i - j + k) = 0$
$\displaystyle 2p - q + r = 0$
So we have the conditions:
$\displaystyle \begin{matrix} x + p = 3 \\ y + q = 2 \\ z + r = -6 \end{matrix}$
and
$\displaystyle \begin{matrix}ex = 2 \\ ey = -1 \\ e = z \end{matrix}$
and
$\displaystyle 2p - q + r = 0$
We have 7 variables in 7 unknowns. It'll take a while, but it should be solvable.
-Dan
Vector $\displaystyle \bold{b}$ is the orthogonal projection of $\displaystyle \bold{a}$ on $\displaystyle \bold{d}$ calculated by
$\displaystyle \bold{b} = \frac{\bold{a}\cdot \bold{d}}{\bold{d}\cdot\bold{d}} \bold{d} = \frac{-2}{6} \bold{d} = \frac{-1}{3} \bold{d}.$
Then set $\displaystyle \bold{c} = \bold{a} - \bold{b}.$
Now check that $\displaystyle \bold{a} = \bold{b} + \bold{c}$ and $\displaystyle \bold{b}$ is parallel to $\displaystyle \bold{d}$ since it is a multiple of it.
Finally $\displaystyle \bold{c}$ is perpendicular to $\displaystyle \bold{d}$ since
$\displaystyle \begin{aligned} \bold{c}\cdot\bold{d} &= (\bold{a} - \bold{b})\cdot\bold{d}.\\
&= \bold{a}\cdot\bold{d} - \bold{b}\cdot\bold{d} \\
&= \bold{a}\cdot\bold{d} - \frac{\bold{a}\cdot \bold{d}}{\bold{d}\cdot\bold{d}} \bold{d}\cdot\bold{d} \\
&= \bold{a}\cdot\bold{d} - \bold{a}\cdot\bold{d} \\
&= 0.
\end{aligned}$