Chapter 10: Q49E (page 557)
Prove the Cauchy-Schwarz Inequality theorem.
The proving is stated below.
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
To show: The expression \(|a \times b{|^2} = |a{|^2}|b{|^2} - {(a \cdot b)^2}\).
(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 unit vector \(\left( a \right).\)
To determine whether the given vectors are orthogonal, parallel, or neither.
(a) For vector\({\rm{a}} = \langle - 5,3,7\rangle \)and\({\rm{b}} = \langle 6, - 8,2\rangle \)
(b) For vector\(a = \langle 4,6\rangle \)and\(b = \langle - 3,2\rangle \)
(c) For vector\({\bf{a}} = - {\bf{i}} + 2{\bf{j}} + 5{\bf{k}}\)and\({\bf{b}} = - 3{\bf{i}} + 4{\bf{j}} - {\bf{k}}\)
(d) For vector\({\bf{a}} = 2{\bf{i}} + 6{\bf{j}} - 4{\bf{k}}\)and\({\bf{b}} = - 3{\bf{i}} - 9{\bf{j}} + 6{\bf{k}}\)
To show
(a) \({\rm{i}} \cdot {\rm{j}} = 0,{\rm{j}} \cdot {\rm{k}} = 0\) and \({\rm{k}} \cdot {\rm{i}} = 0\).
(b) \({\rm{i}} \cdot {\rm{i}} = 1,{\rm{j}} \cdot {\rm{j}} = 1\) and \({\rm{k}} \cdot {\rm{k}} = 1\)
What do you think about this solution?
We value your feedback to improve our textbook solutions.
