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(a)

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

(b)

\\[\\boldsymbol{r} = \\var{vector(a1,b1,g1)} + \\mu \\var{vector(c1,d1,f1)}\\]

(c)

Since the two lines intersect at $\\boldsymbol{P}$, we must have that

\\[ \\var{vector(a,b,g)} + \\lambda \\var{vector(c,d,f)} = \\var{vector(a1,b1,g1)} + \\mu \\var{vector(c1,d1,f1)}\\]

at point $\\boldsymbol{P}$

Hence we can write out a set of simultaneous equations for each component of $\\boldsymbol{P}$:

\\begin{align}
\\simplify[]{{a} + lambda*{c}} &= \\simplify[]{{a1} + mu*{c1}} \\\\
\\simplify[]{{b} + lambda*{d}} &= \\simplify[]{{b1} + mu*{d1}} \\\\
\\simplify[]{{g} + lambda*{f}} &= \\simplify[]{{g1} + mu*{f1}}
\\end{align}

By solving these equations, we find that the point $\\boldsymbol{P}$ common to both lines is given by $\\lambda=\\var{lam},\\mu=\\var{mu}$, and

\\[\\boldsymbol{P} = \\var{p}\\]

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Point of intersection of the two lines

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Given two 3 dim vectors, find vector equation of line through one vector in the direction of another. Find two such lines and their point of intersection.

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Find the vector equation of Line 1 which goes through the point $\\boldsymbol{x_0}$ in the direction of the vector $\\boldsymbol{v}$.

Input the vector equation in the form:

\\[\\boldsymbol{r} = \\begin{pmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{pmatrix} + \\lambda \\begin{pmatrix} b_1 \\\\ b_2 \\\\ b_3 \\end{pmatrix} \\]

such that $\\boldsymbol{r} = \\boldsymbol{x_0}$ when $\\lambda=0$ and $\\boldsymbol{r}=\\boldsymbol{x_0}+\\boldsymbol{v}$ when $\\lambda=1$ by filling in the appropriate fields below:

$ \\boldsymbol{r} = $ [[0]] $ + \\lambda $ [[1]]

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Once again find the vector equation of Line 2 which goes through the point $\\boldsymbol{y_0}$ in the direction of the vector $\\boldsymbol{w}$ in the form

\\[ \\boldsymbol{r} = \\begin{pmatrix} c_1 \\\\ c_2 \\\\ c_3 \\end{pmatrix} + \\mu \\begin{pmatrix} d_1 \\\\ d_2 \\\\ d_3 \\end{pmatrix} \\]

such that $\\boldsymbol{r}=\\boldsymbol{C}$ when $\\mu=0$ and $\\boldsymbol{r=C+D}$ when $\\mu=1$ by filling in the appropriate fields below:

$ \\boldsymbol{r} = $ [[0]] $ + \\mu $ [[1]]

You are told that Line 1 and Line 2 intersect in a point $\\boldsymbol{P}$.

Find $\\boldsymbol{P}$.

$\\boldsymbol{P} = $ [[0]]

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You are given the vectors

\\begin{align}
\\boldsymbol{x_0} &= \\var{vector(a,b,g)} , & \\boldsymbol{v} & = \\var{vector(c,d,f)}, \\\\[1em]
\\boldsymbol{y_0} &= \\var{vector(a1,b1,g1)}, & \\boldsymbol{w} &=\\var{vector(c1,d1,f1)}
\\end{align}

in $\\mathbb{R^3}$.

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