Given two vectors in R4 find two vectors satisfying given relations

Let $A = (1,2,3,4)$ and $B = (1,1,1,1)$ . Find vectors $P,Q$ such that $A = P+Q$ , $P$ is parallel to $B$ , and $Q$ is orthogonal to $B$ .

Let $P = (p_1, p_2, p_3, p_4)$ and $Q = (q_1, q_2, q_3, q_4)$ . Then since $A = P+Q$ we have

$\begin{align*} p_1 + q_1 &= 1 \\ p_2 + q_2 &= 2 \\ p_3 + q_3 &= 3 \\ p_4 + q_4 &= 4. \end{align*}$

Since $P$ is parallel to $B$ we have

$p_1 = p_2 = p_3 = p_4 = c$

for some nonzero scalar $c$ . Since $Q$ is orthogonal to $B$ we have

$q_1 + q_2 + q_3 + q_4 = 0.$

Solving the first set of equations we have

$\begin{align*} q_1 &= 1 - c\\ q_2 &= 2 - c \\ q_3 &= 3 - c \\ q_4 &= 4 - c. \end{align*}$

Then from the orthogonality we have

$(1-c) + (2-c) + (3-c) + (4-c) = 0 \quad \implies \quad c = \frac{5}{2}.$

Therefore,

$P = \frac{5}{2}(1,1,1,1), \qquad Q = \frac{1}{2}(-3,-1,1,3).$

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