91Ó°ÊÓ

Determine whether the following set of vectors is orthogonal. If it is orthogonal, determine whether it is also orthonormal. $$ \left[\begin{array}{c} \frac{1}{6} \sqrt{2} \sqrt{3} \\ \frac{1}{3} \sqrt{2} \sqrt{3} \\ -\frac{1}{6} \sqrt{2} \sqrt{3} \end{array}\right],\left[\begin{array}{c} \frac{1}{2} \sqrt{2} \\ 0 \\ \frac{1}{2} \sqrt{2} \end{array}\right],\left[\begin{array}{c} -\frac{1}{3} \sqrt{3} \\ \frac{1}{3} \sqrt{3} \\ \frac{1}{3} \sqrt{3} \end{array}\right] $$ If the set of vectors is orthogonal but not orthonormal, give an orthonormal set of vectors which has the same span.

Here are some vectors. $$ \left[\begin{array}{r} 1 \\ 1 \\ -2 \end{array}\right],\left[\begin{array}{r} 1 \\ 2 \\ -2 \end{array}\right],\left[\begin{array}{r} 1 \\ -3 \\ -2 \end{array}\right],\left[\begin{array}{r} -1 \\ 1 \\ 2 \end{array}\right] $$ Now here is another vector: $$ \left[\begin{array}{r} 1 \\ 2 \\ -1 \end{array}\right] $$ Is this vector in the span of the first four vectors? If it is, exhibit a linear combination of the first four vectors which equals this vector, using as few vectors as possible in the linear combination.

Determine whether the following set of vectors is orthogonal. If it is orthogonal, determine whether it is also orthonormal. $$ \left[\begin{array}{l} 1 \\ 0 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{l} 0 \\ 0 \\ 0 \\ 1 \end{array}\right] $$ If the set of vectors is orthogonal but not orthonormal, give an orthonormal set of vectors which has the same span.

For \(U\) an orthogonal matrix, explain why \(\|U \vec{x}\|=\|\vec{x}\|\) for any vector \(\vec{x} .\) Next explain why if \(U\) is an \(n \times n\) matrix with the property that \(\|U \vec{x}\|=\|\vec{x}\|\) for all vectors, \(\vec{x},\) then \(U\) must be orthogonal. Thus the orthogonal matrices are exactly those which preserve length.

The wind blows from West to East at a speed of 50 miles per hour and an airplane which travels at 400 miles per hour in still air heading somewhat West of North so that, with the wind, it is flying due North. It uses 30.0 gallons of gas every hour. If it has to travel 600.0 miles due North, how much gas will it use in flying to its destination?

Are the following vectors linearly independent? If they are, explain why and if they are not, exhibit one of them as a linear combination of the others. Also give a linearly independent set of vectors which has the same span as the given vectors. $$ \left[\begin{array}{r} -1 \\ -2 \\ 2 \\ 3 \end{array}\right],\left[\begin{array}{r} -3 \\ -4 \\ 3 \\ 3 \end{array}\right],\left[\begin{array}{r} 0 \\ -1 \\ 4 \\ 3 \end{array}\right],\left[\begin{array}{r} 0 \\ -1 \\ 6 \\ 4 \end{array}\right] $$

A bird flies from its nest \(8 \mathrm{~km}\) in the direction \(\frac{5}{6} \pi\) north of east where it stops to rest on a tree. It then flies \(1 \mathrm{~km}\) in the direction due southeast and lands atop a telephone pole. Place an \(x y\) coordinate system so that the origin is the bird's nest, and the positive \(x\) axis points east and the positive \(y\) axis points north. Find the displacement vector from the nest to the telephone pole.

Here are some vectors in \(\mathbb{R}^{4}\). $$ \left[\begin{array}{r} 1 \\ 2 \\ -2 \\ 1 \end{array}\right],\left[\begin{array}{r} 1 \\ 3 \\ -3 \\ 1 \end{array}\right],\left[\begin{array}{r} 1 \\ -1 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{r} 2 \\ -3 \\ 3 \\ 2 \end{array}\right],\left[\begin{array}{r} 1 \\ 3 \\ -2 \\ 1 \end{array}\right] $$ Thse vectors can't possibly be linearly independent. Tell why. Next obtain a linearly independent subset of these vectors which has the same span as these vectors. In other words, find a basis for the span of these vectors.

Let \(H\) denote span \(\left\\{\left[\begin{array}{r}-2 \\ 1 \\ 1 \\\ -3\end{array}\right],\left[\begin{array}{r}-9 \\ 4 \\ 3 \\\ -9\end{array}\right],\left[\begin{array}{r}-33 \\ 15 \\ 12 \\\ -36\end{array}\right],\left[\begin{array}{r}-22 \\ 10 \\ 8 \\\ -24\end{array}\right]\right\\}\). Find the dimension of \(H\) and determine a basis.