7.4: The Cross Product
- Page ID
- 143597
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
- Calculate the determinant of \(\begin{bmatrix}
-4 & 1 \\
-2 & -1 \\
\end{bmatrix}\).
- Calculate the determinant of \(\begin{bmatrix}
3 & -2 \\
-6 & 4 \\
\end{bmatrix}\).
- Calculate the determinant of \(\begin{bmatrix}
-1 & 2 & 1 \\
-3 & -3 & -3 \\
-1 & -1 & -3
\end{bmatrix}\).
- Calculate the determinant of \(\begin{bmatrix}
2 & 0 & 2 \\
2 & -3 & -3 \\
-1 & -3 & 1
\end{bmatrix}\).
- Calculate \(\mathbf{u} \times \mathbf{v}\) for \(\mathbf{u} = (1, 3, -3)\) and \(\mathbf{v} = (2, 2, 3)\). Express your answer in component form.
- Calculate \(\mathbf{a} \times \mathbf{b}\) for \(\mathbf{a} = 5\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}\) and \(\mathbf{b} = 4\mathbf{i} + 3\mathbf{k}\). Express your answer in terms of the standard unit vectors.
- Let \(\mathbf{p} = (0, -4, -2)\) and \(\mathbf{q} = (5, 4, 2)\). Find a unit vector \(\mathbf{r}\) pointing in the same direction as \(\mathbf{p} \times \mathbf{q}\). Express your answer in component form.
- Let \(\mathbf{v} = -\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}\) and \(\mathbf{w} = 2\mathbf{i} - 4\mathbf{k}\). Find a unit vector \(\mathbf{u}\) pointing in the same direction as \(\mathbf{v} \times \mathbf{w}\). Express you answer in terms of the standard unit vectors.
- Let \(\mathbf{x} = (3, 5, -1)\) and \(\mathbf{y} = (-1, 3, -4)\), and let \(\theta \in [0, \pi]\) be the angle between \(\mathbf{x}\) and \(\mathbf{y}\). Calculate \(\sin \theta\) and \(\cos \theta\).
- Let \(\mathbf{a} = -4(\mathbf{i} + \mathbf{j})\) and \(\mathbf{b} = 3\mathbf{i} + \mathbf{j} - 5\mathbf{k}\), and let \(\theta \in [0, \pi]\) be the angle between \(\mathbf{a}\) and \(\mathbf{b}\). Calculate \(\sin \theta\) and \(\cos \theta\).
- For \(\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{R}^3\), is \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\) a well-defined expression? Why or why not?
- For \(\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{R}^3\), is \((\mathbf{u} \cdot \mathbf{v}) \times \mathbf{w}\) a well-defined expression? Why or why not?
- For \(\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{R}^3\), is \(\mathbf{u} \times \mathbf{v} \times \mathbf{w}\) a well-defined expression? Why or why not?
- Consider points \(A(0, 3, 2)\), \(B(1, 1, -1)\), and \(C(2, -1, 0)\). Find the area of the parallelogram with adjacent sides \(\vec{AB}\) and \(\vec{AC}\).
- Consider points \(P(2, 3, 2)\), \(Q(-2, -3, 1)\), and \(R(2, 1, 1)\). Find the area of the parallelogram with adjacent sides \(\vec{PQ}\) and \(\vec{PR}\).
- Consider points \(R(0, 2)\), \(S(-1, 1)\), and \(T(-2, 3)\). Find the area of the parallelogram with adjacent sides \(\vec{RS}\) and \(\vec{RT}\).
- Find the volume of the parallelepiped with adjacent edges \(\mathbf{u} = (0, 1, 1)\), \(\mathbf{v} = (1, 2, -2)\), and \(\mathbf{w} = (0, 2, 3)\).
- Find the volume of the parallelepiped with adjacent edges \(\mathbf{a} = \mathbf{i} - 3\mathbf{j} - 2\mathbf{k}\), \(\mathbf{b} = 2\mathbf{i} - \mathbf{j} + 2\mathbf{k}\), and \(\mathbf{c} = 3(\mathbf{i} + \mathbf{j} - \mathbf{k})\).