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Elementary operations on vectors; sum, modulus, unit vector, scalar multiple. 

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(a)

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\\[\\boldsymbol{v}+\\boldsymbol{w} = \\var{vector(a,b,g)} + \\var{vector(c,d,f)} = \\var{vector(a+c,b+d,g+f)} \\]

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(b)

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\\begin{align}
\\var{a4}\\boldsymbol{v} &= \\var{a4}\\times \\var{vector(a,b,g)}
&= \\var{a4*vector(a,b,g)}
\\end{align}

\n

\\begin{align}
\\var{b4}\\boldsymbol{w} &= \\var{b4}\\times\\var{vector(c, d, f)}
&= \\var{b4*vector(c,d,f)}
\\end{align}

\n

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(c)

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$4\\mathbf{v}-2\\mathbf{w} = 4\\times \\var{vector(a,b,g)}-2\\times \\var{vector(c,d,f)}=\\var{vector(4*a,4*b,4*g)}-\\var{vector(2*c,2*d,2*f)}= \\var{vector(4*a-2*c,4*b-2*d,4*g-2*f)}$

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(d)

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In general for a vector $\\boldsymbol{x}= \\begin{pmatrix}x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix}$, we have $\\lVert \\boldsymbol{x} \\rVert = \\sqrt{x_1^2+x_2^2+x_3^2}$.

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Hence:

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\\begin{align}
\\lVert \\boldsymbol{v} \\rVert &= \\sqrt{\\var{a^2}+\\var{b^2}+\\var{g^2}} = \\simplify[all]{ sqrt({a^2+b^2+g^2})} \\\\
\\lVert \\boldsymbol{w} \\rVert &= \\sqrt{\\var{c^2}+\\var{d^2}+\\var{f^2}} = \\simplify[all]{ sqrt({c^2+d^2+f^2})} \\\\
\\lVert \\boldsymbol{v+w} \\rVert &= \\sqrt{\\var{(a+c)^2}+\\var{(b+d)^2}+\\var{(g+f)^2}} = \\simplify[all]{ sqrt({(a+c)^2+(b+d)^2+(f+g)^2})}
\\end{align}

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(e)

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Given a vector $\\boldsymbol{z}= \\begin{pmatrix} z_1 \\\\ z_2 \\\\ z_3 \\end{pmatrix}$, the unit vector parallel to $\\boldsymbol{z}$ is given by:

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\\[ \\boldsymbol{\\hat{z}} = \\frac{1}{\\lVert \\boldsymbol{z} \\rVert} \\begin{pmatrix} z_1 \\\\ z_2 \\\\ z_3 \\end{pmatrix} = \\begin{pmatrix} \\frac{z_1}{\\lVert \\boldsymbol{z} \\rVert} \\\\ \\frac{z_2}{\\lVert \\boldsymbol{z} \\rVert} \\\\ \\frac{z_3}{\\lVert \\boldsymbol{z} \\rVert} \\end{pmatrix} \\]

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For this example we have

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$\\boldsymbol{v}+\\boldsymbol{w} = \\var{vector(a+c,b+d,g+f)}$       from part (a)

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and

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$\\lVert \\boldsymbol{v+w} \\rVert =\\simplify[std]{sqrt({(a+c)^2+(b+d)^2+(f+g)^2})}$        from part (d)

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Hence:

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\\begin{align}
&& \\boldsymbol{\\hat{z}} &= \\frac{1}{\\sqrt{\\var{ssquares}}} \\begin{pmatrix} \\var{a+c} \\\\ \\var{b+d} \\\\ \\var{g+f} \\end{pmatrix} \\\\[1em] 
&& &= \\begin{pmatrix} \\simplify[std]{{a+c}/sqrt({ssquares})} \\\\ \\simplify[std]{{b+d}/sqrt({ssquares})} \\\\ \\simplify[std]{{g+f}/sqrt({ssquares})} \\end{pmatrix}
\\end{align}

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You are given the vectors

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\\begin{align}
\\mathbf{v} & =\\simplify[std]{vector({a},{b},{g})}, &
\\mathbf{w} &= \\simplify[std]{vector({c},{d},{f})}\\qquad \\in{\\mathbb R}^3.
\\end{align}

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Enter your answers to the following questions exactly, using the function sqrt(x) if necessary.

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Calculate $\\mathbf{v}+\\mathbf{w} = $ [[0]]

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Calculate

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$\\var{a4}\\mathbf{v} = $ [[0]]

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$\\var{b4}\\mathbf{w} = $ [[1]]

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Calculate $4\\mathbf{v}-2\\mathbf{w} = $[[0]]

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Calculate the following.

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$\\vert \\mathbf{v} \\vert=$ [[0]]

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$\\vert \\mathbf{w} \\vert = $ [[1]]

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$\\vert \\mathbf{v}+\\mathbf{w} \\vert = $ [[2]]

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You must enter your answers exactly, using the function sqrt(x) as necessary

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Let $\\mathbf{z}=\\mathbf{v}+\\mathbf{w}$.

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Calculate the unit vector $\\mathbf{\\hat{z}}$ in the direction of $\\mathbf{z}$. Write $\\mathbf{\\hat{z}}$ as a row vector.

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$\\mathbf{\\hat{z}}= \\big($ [[0]], [[1]], [[2]] $\\big)$

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You must enter your answers exactly, using the function sqrt(x) as necessary.

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