Chapter 10: Q78E (page 591)
Find r(t)=u(t) x v(t), where u and v are the vector functions in Exercise 77, find r’(2).
\(r'(2) = \left\langle {12, - 29,14} \right\rangle \)
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(a) To determine
To find: A nonzero vector orthogonal to the plane through the points \({\bf{P}}\), \({\bf{Q}}\) and \(R\).
(b) To determine
To find: The area of triangle \({\bf{PQ}}R\).
To determine the meaning of the dot product \({\rm{A}} \cdot {\rm{P}}\).
(a) Let \(P\) be a point not on the line \(L\) that passes through the points \(Q\) and \(R\). Show that the distance \(d\) from the point \(P\) to the line \(L\) is
\(d = \frac{{|{\bf{a}} \times {\bf{b}}|}}{{|{\bf{a}}|}}\)
where \({\bf{a}} = \overrightarrow {QR} \) and \({\bf{b}} = \overrightarrow {QP} \).
(b) Use the formula in part (a) to find the distance from the point \(P(1,1,1)\) to the line through \(Q(0,6,8)\) and \(R( - 1,4,7)\).
(a) Determine the vector \({{\rm{k}}_i}\) is perpendicular to \({{\rm{v}}_j}\) except at \(i = j\).
(b) Determine the dot product \({{\rm{k}}_i} \cdot {{\rm{v}}_i} = 1\).
(c) Determine the condition \({{\rm{k}}_1} \cdot \left( {{{\rm{k}}_2} \times {{\rm{k}}_3}} \right) = \frac{1}{{{{\rm{v}}_1} \cdot \left( {{{\rm{v}}_2} \times {{\rm{v}}_3}} \right)}}\).
(a) To find the parallel unit vectors to the tangent line of \(y = 2\sin x\).
(b) To find the perpendicular unit vectors to the tangent line of \(y = 2\sin x\).
(c) To sketch curve of \(y = 2\sin x\) along with vectors \( \pm \frac{1}{2}({\bf{i}} + \sqrt 3 {\bf{j}})\) and \( \pm \frac{1}{2}(\sqrt 3 {\bf{i}} - {\bf{j}})\).
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