91Ó°ÊÓ

Find the cross product of the unit vectors [where \(\mathbf{i}=(1,0,0), \mathbf{j}=(0,1,0), \text { and } \mathbf{k}=(0,0,1)] .\) Sketch your result. $$\mathbf{i} \times \mathbf{j}$$

Find the cross product of the unit vectors [where \(\mathbf{i}=(1,0,0), \mathbf{j}=(0,1,0), \text { and } \mathbf{k}=(0,0,1)] .\) Sketch your result. $$\mathbf{j} \times \mathbf{k}$$

Find the orthogonal complement \(S^{\perp}\). \(S\) is the subspace of \(R^{5}\) consisting of all vectors whose third and fourth components are zero.

Determine whether the set of vectors in \(R^{n}\) is orthogonal, orthonormal, or neither. $$\\{(2,-4,2),(0,2,4),(-10,-4,2)\\}$$

Find the orthogonal projection of \(\mathbf{b}=\left[\begin{array}{lll}2 & -2 & 1\end{array}\right]^{T}\) onto the column space of the matrix \(A\). $$A=\left[\begin{array}{ll} 0 & 2 \\ 1 & 1 \\ 1 & 3 \end{array}\right]$$

Find bases for the four fundamental subspaces of the matrix \(A\). $$A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 0 & 1 & 0 \end{array}\right]$$

Find the least squares regression line for the data points. Graph the points and the line on the same set of axes. $$(-2,-1),(-1,0),(1,0),(2,2)$$

Find the least squares regression line for the data points. Graph the points and the line on the same set of axes. $$(-2,1),(-1,2),(0,1),(1,2),(2,1)$$

Find the least squares regression line for the data points. Graph the points and the line on the same set of axes. $$(-2,0),(-1,2),(0,3),(1,5),(2,6)$$

Show that the volume of a parallelepiped having \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) as adjacent sides is the triple scalar product \(|\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})|\).