Chapter 10: Q43E (page 590)
Find the derivative of the vector function
r ( t ) = a + tb + t2 c
The derivative of the vector function is
\[r'(t) = b + 2tc\]
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To d\({\bf{b}} = \left\langle {{b_1},{b_2},{b_3}} \right\rangle \)escribe all set of points for condition \(\left| {{\bf{r}} - {{\bf{r}}_0}} \right| = 1\).
(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 \({\bf{PQ}}R\).
(a) Examine if\(a \cdot b = a \cdot c\), does it follow that\(b = c\).
(b) Examine if\(a \times b = a \times c\), does it follow that\(b = c\).
(c) Examine if\(a \cdot b = a \cdot c\)and\(a \times b = a \times c\), does it follow that\(b = c\).
(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 \(R( - 1,4,7)\).
To show: The expression \(|a \times b{|^2} = |a{|^2}|b{|^2} - {(a \cdot b)^2}\).
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