Verify the Cauchy-Schwarz inequality and the triangle inequality for \(\u=\left(\begin{array}{r}2\\-2\\1 \end{array} \right),\,\v=\left(\begin{array}{r}-4\\3\\12 \end{array} \right)\)Hint.
Consider the LHS and RHS of each inequality and by inspection determine if the relation in each of the Cauchy-Schwarz and triangle inequalities holds true.
Exercise Group.
Find vectors \(\begin{pmatrix}x\\y\\z \end{pmatrix}\) which are orthogonal to \(\left(\begin{array}{r}-1\\2\\-3 \end{array} \right),\) as follows.
2.
Find two such vectors with \(0\)’s for the first component
3.
Find two such vectors with \(0\)’s for the second component
4.
Find two such vectors with \(0\)’s for the third component
5.
Find two such vectors with no \(0\)’s
6.
Can you find such a vector with two \(0\)’s? Explain.
7.
Prove that no nonzero vector is orthogonal to iself.
Find a one-dimensional (one-parameter) infinite set of vectors orthogonal to \(\left(\begin{array}{r} 5\\4\\-1 \end{array} \right),\) as follows.
9.
In demanding that a vector \(\begin{pmatrix}x\\y\\z \end{pmatrix}\) to be orthogonal to \(\left(\begin{array}{r} 5\\4\\-1 \end{array} \right)\) we obtain one equation in \(x,y,z.\) What is that equation?
10.
Fix \(y=0,\) and solve the equation above for \(z\) in terms of \(x.\)
11.
Using your result from Worksheet Exercise 2.5.1.10, write the general form of a vector with \(y=0\) which is orthogonal to \(\left(\begin{array}{r} 5\\4\\-1 \end{array} \right):\)
12.
Fix \(y=1,\) and repeat the last two steps above.
13.
Fix \(z=1,\) and repeat the last two steps above, with appropriate modifications.
Exercise Group.
Find a two-dimensional infinite set of vectors orthogonal to \(\left(\begin{array}{r}5\\3\\4 \end{array} \right),\) as follows.
14.
In demanding that a vector \(\left(\begin{array}{r} x\\y\\z \end{array} \right)\) to be orthogonal to \(\left(\begin{array}{r}5\\3\\4 \end{array} \right)\) we obtain one equation in \(x,y,z.\) What is that equation?
15.
Solve the equation above for \(z\) in terms of \(x,y.\)
16.
From this, write the general form of a vector which is orthogonal to \(\left(\begin{array}{r}5\\3\\4 \end{array} \right):\)
17.
Find three mutually orthogonal vectors in \(\R^4.\)
18.
Find two vectors in \(\R^3\) which are neither parallel nor anti-parallel to each other and lie at an angle of \(\displaystyle{\theta=\frac{\pi}{3}}\) to the vector \(\v=\left(\begin{array}{r}1\\1\\1 \end{array} \right).\)
Hint.
Let \(\u=\begin{pmatrix}x\\y\\z \end{pmatrix}\) and write out the equation you get from \(\dpr{\u}{\v}=\|\u\|\,\|\v\|\cos\th.\) Let \(y=1=-z\) for one of the solutions (watch the extraneous roots!) and let \(y=-1=-z\) for the other solution. Be sure to check the non-parallel and non-anti-parallel conditions are satisfied.
Let \(\u,\v,\w\in\R^n\) with \(\u\perp\w\) and \(\v\perp\w;\) let \(a,b\in\R.\) Determine, with proof, whether the linear combination \((a\u+b\v)\perp\w.\)
if and only if \(\u\parallel\v\) (in which case \(\u\cdot\v=\|\u\|\|\v\|\)) or \(\u\upharpoonleft\!\downharpoonright\v\) (in which case \(\u\cdot\v=-\|\u\|\|\v\|\)).
