Specialist Mathematics · Unit 3 · Vectors in two and three dimensions · Algebra of vectors in three dimensions
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Two forces acting on an object are represented by the vectors '\(\vec{F_1} = 3\mathbf{i} - 2\mathbf{j} + \mathbf{k}\)' N and '\(\vec{F_2} = \mathbf{i} + 4\mathbf{j} - \mathbf{k}\)' N. (a) Calculate the scalar product '\(\vec{F_1} \cdot \vec{F_2}\)'. (1 mark) (b) Hence, determine whether the two forces are perpendicular. Justify your answer. (1 mark)
Three position vectors are given in the table below. Use the scalar product to determine the value of \(k\) for which \(\vec{OA}\) is perpendicular to \((\vec{OB} + \vec{OC})\).
Given \(\vec{a} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k}\), \(\vec{b} = -\mathbf{i} + 4\mathbf{j} + 2\mathbf{k}\), and \(\vec{c} = 3\mathbf{i} + m\mathbf{j} + 5\mathbf{k}\), where \(m \in \mathbb{R}\), determine the value of \(m\) such that \(\vec{c}\) is perpendicular to \((\vec{a} + \vec{b})\).