Chapter 10: Q27E (page 542)
Describe the representation of inequality \({x^2} + {y^2} + {z^2} \le 3\) in \({R^3}\).
The inequality \({x^2} + {y^2} + {z^2} \le 3\) in \({R^3}\) represents a region in which all points lie on or inside the sphere with radius \(\sqrt 3 \) and center is at the origin \((0,0,0)\).
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To determine whether the given vectors are orthogonal, parallel, or neither.
(a)For vector\(u = \langle - 3,9,6\rangle \)and\(v = \langle 4, - 12, - 8\rangle \)
(b)For vector\(u = \langle 1, - 1,2\rangle \)and\(v = \langle 2, - 1,1\rangle \)
(c)For vector\({\rm{u}} = \langle a,b,c\rangle \)and\({\rm{v}} = \langle - b,a,0\rangle \)
Find a vector equation for a line through the point \((2,2.4,3.5)\) and parallel to the vector \(3i + 2j - k\) and the parametric equations for a line through the point \((2,2.4,3.5)\) and parallel to the vector \(3i + 2j - k\).
If \({\bf{a}} + {\bf{b}} + {\bf{c}} = {\bf{0}}\), show that
\({\bf{a}} \times {\bf{b}} = {\bf{b}} \times {\bf{c}} = {\bf{c}} \times {\bf{a}}\)
(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 \({\bf{PQ}}R\).
Show the \(a \times (b \times c) \ne (a \times b) \times c\).
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