7.3: The Dot Product
- Page ID
- 143596
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Let vector \(\mathbf{u} = (2, -3)\) and vector \(\mathbf{v} = (1, -4)\). Find \(\mathbf{u} \cdot \mathbf{v}\).
- Let vector \(\mathbf{a} = 3\mathbf{i} - 7\mathbf{j}\) and vector \(\mathbf{b} = -2\mathbf{j}\). Find \(\mathbf{a} \cdot \mathbf{b}\).
- Let vector \(\mathbf{v} = (-2, -1, 5)\) and vector \(\mathbf{w} = (2, 2, -4)\). Find \(\mathbf{v} \cdot \mathbf{w}\).
- Let vector \(\mathbf{p} = 2\mathbf{j} - 5\mathbf{k}\) and vector \(\mathbf{q} = -4\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}\). Find \(\mathbf{p} \cdot \mathbf{q}\).
- Let vector \(\mathbf{u} = (-2, 4)\), vector \(\mathbf{v} = (-5, 5)\), and vector \(\mathbf{w} = (-1, 0)\). Evaluate \((\mathbf{u} \cdot \mathbf{v})\mathbf{w}\), if possible. If it is not possible, explain why not.
- Let vector \(\mathbf{a} = 3\mathbf{i} + 4\mathbf{j}\), vector \(\mathbf{b} = -2\mathbf{i} - 5\mathbf{j}\), and vector \(\mathbf{c} = \mathbf{i} - 5\mathbf{j}\). Evaluate \(\mathbf{a} \cdot \mathbf{b} \cdot \mathbf{c}\), if possible. If it is not possible, explain why not.
- Let vector \(\mathbf{r} = (3, 3, -1)\), vector \(\mathbf{s} = (0, 5, -3)\), and vector \(\mathbf{t} = (3, 0, 2)\). Evaluate \((\mathbf{r} \cdot \mathbf{s}) + \mathbf{t}\), if possible. If it is not possible, explain why not.
- Let vector \(\mathbf{x} = 2\mathbf{i} - \mathbf{k}\), vector \(\mathbf{y} = 2\mathbf{i} + 2\mathbf{j}\), and vector \(\mathbf{z} = -5\mathbf{i} - 5\mathbf{j} - 5\mathbf{k}\). Evaluate \((\mathbf{x} + \mathbf{y}) \cdot \mathbf{z}\), if possible. If it is not possible, explain why not.
- Let \(\theta \in [0, \pi]\) be the angle between two vectors \(\mathbf{u}\) and \(\mathbf{v}\). Recalling that \(\mathbf{u} \cdot \mathbf{v} = \| \mathbf{u}\| \| \mathbf{v}\| \cos \theta\), what can you say about \(\theta\) if \( \mathbf{u} \cdot \mathbf{v} = 0\)? What can you say about \(\theta\) if \( \mathbf{u} \cdot \mathbf{v} > 0\)? What can you say about \(\theta\) if \( \mathbf{u} \cdot \mathbf{v} < 0\)?
- Find the measure of the angle between vector \(\mathbf{u} = (-1, 1, 2)\) and vector \(\mathbf{v} = (-3, 1, -2)\).
- Find the measure of the angle between vector \(\mathbf{a} = \mathbf{i} + \mathbf{j}\) and vector \(\mathbf{b} = \mathbf{j} - \mathbf{k}\).
- Find the measure of the angle between vector \(\mathbf{v} = (1, -2, 1)\) and vector \(\mathbf{w} = (1, 1, -2)\).
- Find a non-zero vector \(\mathbf{p} \in \mathbb{R}^2\) that is orthogonal to \(\mathbf{q} = 2\mathbf{i} - 3\mathbf{j}\). Does there exist another non-zero vector \(\mathbf{r} \in \mathbb{R}^2\), not parallel to \(\mathbf{p}\), that is also orthogonal to \(\mathbf{q}\)?
- Find a non-zero vector \(\mathbf{x} \in \mathbb{R}^3\) that is orthogonal to \(\mathbf{y} = (4, 1, -2)\). Does there exist another non-zero vector \(\mathbf{z} \in \mathbb{R}^3\), not parallel to \(\mathbf{x}\), that is also orthogonal to \(\mathbf{y}\)?
- Let vector \(\mathbf{a} = 4\mathbf{i} + 5\mathbf{j}\) and vector \(\mathbf{b} = -\mathbf{i} - 2\mathbf{j}\). Find \(\text{proj}_{\mathbf{b}} \mathbf{a}\), the projection of vector \(\mathbf{a}\) onto vector \(\mathbf{b}\).
- Let vector \(\mathbf{u} = (2, 2)\) and vector \(\mathbf{v} = (-5, 3)\). Decompose vector \(\mathbf{v}\) into the sum of a vector parallel to \(\mathbf{u}\) and a vector orthogonal to \(\mathbf{u}\).
- Let vector \(\mathbf{p} = -3\mathbf{i} + 2\mathbf{j} - \mathbf{k}\) and vector \(\mathbf{q} = 3\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}\). Decompose vector \(\mathbf{p}\) into the sum of a vector parallel to \(\mathbf{q}\) and a vector orthogonal to \(\mathbf{q}\).