1 problem found
The vectors \({\bf a}\) and \({\bf b}\) lie in the plane \(\Pi\,\). Given that \(\vert {\bf a} \vert= 1\) and \({\bf a}.{\bf b} = 3,\) find, in terms of \({\bf a}\) and \({\bf b}\,\), a vector \({\bf p}\) parallel to \({\bf a}\) and a vector \({\bf q}\) perpendicular to \({\bf a}\,\), both lying in the plane \(\Pi\,\), such that $${\bf p}+{\bf q}={\bf a}+{\bf b}\;.$$ The vector \({\bf c}\) is not parallel to the plane \(\Pi\) and is such that \({\bf a}.{\bf c} = -2\) and \({\bf b}.{\bf c} = 2\,\). Given that \(\vert {\bf b} \vert = 5\,\), find, in terms of \({\bf a}, {\bf b}\) and \({\bf c},\) vectors \({\bf P}\), \({\bf Q}\) and \({\bf R}\) such that \({\bf P}\) and \({\bf Q}\) are parallel to \({\bf p}\) and \({\bf q},\) respectively, \({\bf R}\) is perpendicular to the plane \(\Pi\) and $${\bf P} + {\bf Q} + {\bf R} = {\bf a}+{\bf b}+{\bf c}\;.$$
Solution: Suppose \({\bf p} = \lambda {\bf a}\) and \({\bf p} + {\bf q} = {\bf a} + {\bf b}\) then \begin{align*} {\bf a} \cdot : && {\bf a} \cdot {\bf p} + {\bf a} \cdot {\bf p} &= {\bf a} \cdot {\bf a} + {\bf a} \cdot {\bf b} \\ && \lambda + 0 &= 1 + 3 = 4 \\ \Rightarrow && \mathbf{p} &= 4 \mathbf{a} \\ && \mathbf{q} &= \mathbf{b} - 3\mathbf{a} \\ \\ && \mathbf{P} &= 4p\mathbf{a} \\ && \mathbf{Q} &= q\mathbf{b} - 3q\mathbf{a} \\ \\ \mathbf{a} \cdot : && \mathbf{a} \cdot \mathbf{P} + \mathbf{a} \cdot \mathbf{Q} + \mathbf{a} \cdot \mathbf{R} &= \mathbf{a} \cdot \mathbf{a} + \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} \\ && 4p &= 1+3-2 \\ \Rightarrow && p &= \tfrac12 \\ \\ && {\bf P} + {\bf Q} + {\bf R} &= {\bf a}+{\bf b}+{\bf c} \\ \mathbf{b} \cdot : && \mathbf{b} \cdot \mathbf{P} + \mathbf{b} \cdot \mathbf{Q} + \mathbf{b} \cdot \mathbf{R} &= \mathbf{b} \cdot \mathbf{a} + \mathbf{b} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} \\ && 12p + 25q - 9q &= 3+25+2 \\ && 6+16q &= 30 \\ \Rightarrow && q &= \tfrac{3}{2}\\ && \\ && \mathbf{P} &= 2\mathbf{a} \\ && \mathbf{Q} &= \tfrac32 \mathbf{b} - \tfrac92 \mathbf{a} \\ && \mathbf{R} &= \tfrac72\mathbf{a} -\tfrac12 \mathbf{b} + \mathbf{c} \end{align*}