91Ó°ÊÓ

Calculate the determinant \(\left|\begin{array}{rrr}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -1 & 7 \\ 2 & 0 & 3\end{array}\right|\).

Calculate the determinant \(\left|\begin{array}{ccc}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 3 & -4 \\ 1 & 6 & -1\end{array}\right|\).

For the following exercises, the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are given. Use determinant notation to find vector \(\mathbf{w}\) orthogonal to vectors \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u}=\left\langle-1,0, e^{t}\right\rangle, \quad \mathbf{v}=\left\langle 1, e^{-t}, 0\right\rangle,\) where \(t\) is a real number

For the following exercises, the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are given. Use determinant notation to find vector \(\mathbf{w}\) orthogonal to vectors \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u}=\langle 1,0, x\rangle, \quad \mathbf{v}=\left\langle\frac{2}{x}, 1,0\right\rangle,\) where \(x\) is a nonzero real number

Find vector \((\mathbf{a}-2 \mathbf{b}) \times \mathbf{c},\) where \(\mathbf{a}=\left|\begin{array}{rr}\mathbf{i} & \mathbf{j} \mathbf{k} \\ 2 & -1 & 5 \\ 0 & 1 & 8\end{array}\right|\), \(\mathbf{b}=\left|\begin{array}{rrr}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 1 & 1 \\ 2 & -1 & -2\end{array}\right|, \quad\) and \(\mathbf{c}=\mathbf{i}+\mathbf{j}+\mathbf{k} .\)

Find vector \(\mathbf{c} \times(\mathbf{a}+3 \mathbf{b}),\) where \(\mathbf{a}=\left|\begin{array}{ccc}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 5 & 0 & 9 \\ 0 & 1 & 0\end{array}\right|\), \(\mathbf{b}=\left|\begin{array}{rrr}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & -1 & 1 \\ 7 & 1 & -1\end{array}\right|, \quad\) and \(\mathbf{c}=\mathbf{i}-\mathbf{k} .\)

[T] Use the cross product \(\mathbf{u} \times \mathbf{v}\) to find the acute angle between vectors \(\mathbf{u}\) and \(\mathbf{v},\) where \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}\) and \(\mathbf{v}=\mathbf{i}+\mathbf{k}\). Express the answer in degrees rounded to the nearest integer.

[T] Use the cross product \(\mathbf{u} \times \mathbf{v}\) to find the obtuse angle between vectors \(\mathbf{u}\) and \(\mathbf{v}\), where \(\mathbf{u}=-\mathbf{i}+3 \mathbf{j}+\mathbf{k}\) and \(\mathbf{v}=\mathbf{i}-2 \mathbf{j}\). Express the answer in degrees rounded to the nearest integer.

Use the sine and cosine of the angle between two nonzero vectors \(\mathbf{u}\) and \(\mathbf{v}\) to prove Lagrange's identity: \(\|\mathbf{u} \times \mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2}-(\mathbf{u} \cdot \mathbf{v})^{2}\)

Verify Lagrange's identity \(\|\mathbf{u} \times \mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2}-(\mathbf{u} \cdot \mathbf{v})^{2}\) for vectors \(\mathbf{u}=-\mathbf{i}+\mathbf{j}-2 \mathbf{k}\) and \(\mathbf{v}=2 \mathbf{i}-\mathbf{j} .\)