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  1. Suppose \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v} = \vc{}{+2}{-2} and \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{w} = \vc{4}{+0}{-3}. Find \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}\cdot\vv{w}, the angle between \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v} and \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{w}, the scalar projection of \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v} onto \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{w}, and \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}\times\vv{w}.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}\cdot\vv{w} = 4 + 0 + 6 = 10

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}|\vv{v}| = \sqrt{1+4+4} = 3
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}|\vv{w}| = \sqrt{16+9} = 5

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\theta_{\vv{v},\vv{w}}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= \arccos{\vv{v}\cdot\vv{w} \over |\vv{v}||\vv{w}|}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= \arccos{10\over(3)(5)} = \arccos{2\over3}

    Scalar projection \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}s_{\vv{v}\rightarrow\vv{w}}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= |\vv{v}|\cos{\theta_{\vv{v},\vv{w}}}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= 3{2\over3} = 2

    \vv{v}\times\vv{w} = \begin{vmatrix} 1 & 2 & -2 \\ 4 & 0 & -3 \\ \vh{i} & \vh{j} & \vh{k} \end{vmatrix}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= \vc{(-6+0)}{-(-3+8)}{+(0-8)}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}~~~~= \vc{-6}{-5}{-8}

  2. Given a vector \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v} = \vc{}{+2}{+0}, find a vector on the xy plane orthogonal to \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{w} = \vc{x}{+y}{+0}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}\cdot\vv{w} = 0
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}x + 2y = 0
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}x = -2y

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{w}_{y=1} = \vc{-2}{+1}{+0}

  3. Which of the following products of vectors makes sense. Justify why or why not for each one: \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}\cdot(\vv{w}\times\vv{u}), \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}\cdot\vv{w}\cdot\vv{u}, \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}\times\vv{w}\times\vv{u}, \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}(\vv{v}\times\vv{w})\times\vv{u}.

    • \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}\cdot(\vv{w}\times\vv{u}):

      Makes sense. \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{w}\times\vv{u} is a vector, as is \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}, so the ‘\newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\cdot‘ is the dot product.

    • \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}\cdot\vv{w}\cdot\vv{u}:

      This could mean \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}(\vv{v}\cdot\vv{w})\cdot\vv{u}, where the second ‘\newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\cdot‘ represents scalar multiplication, but usually this would be written \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}(\vv{v}\cdot\vv{w})\vv{u}.

    • \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}\times\vv{w}\times\vv{u}:

      This could mean \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}(\vv{v}\times\vv{w})\times\vv{u}, but usually this would be written \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}(\vv{v}\times\vv{w})\times\vv{u}. See below.

    • \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}(\vv{v}\times\vv{w})\times\vv{u}:

      Makes sense. \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}\times\vv{w} is a vector, as is \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{u}, so the second ‘\newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\times‘ is the cross product.

  4. Find the parametric equations for a line connecting the points \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}(0,0,0) and \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}(5,1,3).

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{x}(t) = \vc{5t}{+t}{+3t}

  5. Find the parametric equations for a line passing through \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}(1,2,3) and parallel to the vector \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v} = \vc{}{+4}{-2}.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{x}(t) = \vc{(t+1)}{+(4t+2)}{+(3t+3)}

  6. Find a normal vector to the plane containing the three points \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}(1,3,2),(-1,-2,-3),(0,1,0). Using this same normal vector, find the equation of the given plane.

    0 = \begin{vmatrix} x & y-1 & z \\ 1 & 2 & 2 \\ -1 & -3 & -3 \end{vmatrix}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{n} = \vc{0}{-1}{-1}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}0 = -y - z + 1

  7. Sketch the curve given by \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{x}(t) = \vc{3\cos(t)}{+\sin(t)}{+t^2}.

    Sketch of curve

  8. Sketch the curve given by \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{x}(t) = \vc{t}{+\sin(t)}{+\cos(t)}.

    Sketch of curve

  9. Assuming the two curves above are position vectors, find the velocity vector and acceleration vector of each.

    1. \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}(t) = \vc{-3\sin(t)}{+\cos(t)}{+2t}

      \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{a}(t) = \vc{-3\cos(t)}{-\sin(t)}{+2}

    2. \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{v}(t) = \vc{}{+\cos(t)}{-\sin(t)}

      \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{a}(t) = \vc{0}{-\sin(t)}{-\cos(t)}

  10. Sketch the vector field given by \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{F}(\vv{x}) = \vc{y}{+2}{+0}. Use at least 6 distinct points.

    Sketch of vector field

  11. Sketch the vector field given by \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\vv{F}(\vv{x}) = \vc{{-x \over (x^2+y^2)^{3/2}}}{{-y \over (x^2+y^2)^{3/2}}}{+0}. Use at least 6 distinct points. (Remark: This is a 2-dimensional form of the famous r-squared force laws of gravity and electrostatics.)

    Sketch of vector field


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