Given two vectors, find two vectors satisfying given relations

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

Let $P = (p_1, p_2)$ and $Q = (q_1, q_2)$ . Then

$A = P+Q \quad \implies \quad p_1 + q_1 = 1, \quad p_2 + q_2 = 2.$

Since $P$ is parallel to $B$ we know there is a nonzero constant $c$ such that

$P = cB \quad \implies \quad p_1 = 3c, \quad p_2 = 4c \quad \implies p_2 = \frac{4}{3} p_1.$

From the fact that $Q$ is orthogonal to $B$ we have

$Q \cdot B = 0 \quad \implies \quad 3q_1 + 4q_2 = 0 \quad \implies \quad q_2 = -\frac{3}{4} q_1.$

So putting these together we have

$\begin{align*} p_1 + q_1 &= 1 & && p_1 &= \frac{33}{25}& && p_2 &= \frac{44}{25}\\ && \implies && && \implies && \\ \frac{4}{3} p_1 - \frac{3}{4} q_1 &= 2 &&& q_1 &= -\frac{8}{25} & && q_2 &= \frac{6}{25} \end{align*}$

Therefore,

$P = \frac{11}{25}(3,4), \qquad Q = \frac{2}{25}(-4,3).$

Stumbling Robot

Given two vectors, find two vectors satisfying given relations

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