Chapter 10: Q23E (page 598)
Use formula 11 to find the curvature
y = tanx
\(k(x) = \frac{{12{x^2}}}{{{{\left( {1 + 16{x^6}} \right)}^{\frac{3}{2}}}}}\)
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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 find a dot product between \({\rm{a}}\) and \({\rm{b}}\).
(a) To sketch the vectors \({\rm{a}} = \langle 3,2\rangle ,b = \langle 2, - 1\rangle \), and \({\rm{c}} = \langle 7,1\rangle \).
(b) To sketch the summation vector\({\bf{c}} = s{\bf{a}} + t{\bf{b}}\).
(c) To estimate the values of\(s\)and\(t\)using sketch.
(d) To find the exact values of\(s\)and\(t\).
To prove the result \(a \times (b \times c) + b \times (c \times a) + c \times (a \times b) = 0\).
(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 \({{\rm{k}}_1} \cdot \left( {{{\rm{k}}_2} \times {{\rm{k}}_3}} \right) = \frac{1}{{{{\rm{v}}_1} \cdot \left( {{{\rm{v}}_2} \times {{\rm{v}}_3}} \right)}}\).
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