# Express the equations of the lines r · (i −j) + 7 = 0,r · (i + 3j) − 5 = 0 in parametric forms and hence find the position vector of their point of intersection.

Question
Forms of linear equations
Express the equations of the lines
r · (i −j) + 7 = 0,r · (i + 3j) − 5 = 0 in parametric forms and hence find the position
vector of their point of intersection.

2020-11-03
Express the equation of the lines
$$\displaystyle{o}{v}{e}{r}\rightarrow{\left\lbrace{r}\right\rbrace}\cdot{\left({w}{i}{d}{e}\hat{{{i}}}-{w}{i}{d}{e}\hat{{{j}}}+{7}={0},{o}{v}{e}{r}\rightarrow{\left\lbrace{r}\right\rbrace}\cdot{\left({w}{i}{d}{e}\hat{{{i}}}+{3}{w}{i}{d}{e}\hat{{{j}}}\right)}-{5}={0}\right.}$$
in parametric form
$$\displaystyle{L}{e}{t}{w}{i}{d}{e}\hat{{{r}}}={x}{w}{i}{d}{e}\hat{{{i}}}+{y}{w}{i}{d}{e}\hat{{{j}}}+{z}{w}{i}{d}{e}\hat{{{k}}}$$
$$\displaystyle{w}{i}{d}{e}\hat{{{r}}}\cdot{\left({w}{i}{d}{e}\hat{{{i}}}-{w}{i}{d}{e}\hat{{{j}}}\right)}+{7}={0}$$
$$\displaystyle{\left({x}{w}{i}{d}{e}\hat{{{i}}}+{y}{w}{i}{d}{e}\hat{{{j}}}+{z}{w}{i}{d}{e}\hat{{{k}}}\right)}{\left({w}{i}{d}{e}\hat{{{i}}}{w}{i}{d}{e}\hat{{{j}}}\right)}+{7}={0}$$
$$\displaystyle{L}\in{e}{L}_{{1}}:-{x}-{y}+{7}={0}$$
$$\displaystyle{A}{n}{d}{w}{i}{d}{e}\hat{{{r}}}\cdot{\left({w}{i}{d}{e}\hat{{{i}}}+{3}{w}{i}{d}{e}\hat{{{j}}}\right)}-{5}={0}$$
$$\displaystyle{\left({x}{w}{i}{d}{e}\hat{{{i}}}+{y}{w}{i}{d}{e}\hat{{{j}}}+{z}{w}{i}{d}{e}\hat{{{k}}}\right)}{\left({w}{i}{d}{e}\hat{{{i}}}+{3}{w}{i}{d}{e}\hat{{{a}{b}}}\right)}-{5}={0}$$
$$\displaystyle{L}\in{e}{L}_{{2}}{x}+{3}{y}-{5}={0}$$
Line $$\displaystyle{L}_{{1}}$$ is in direction of $$\displaystyle{w}{i}{d}{e}\hat{{{i}}}-{w}{i}{d}{e}\hat{{{j}}}--{\left({v}\right)}$$
And $$\displaystyle{L}_{{1}}$$ also passes through (-3,4)(A)
Vector equation of $$\displaystyle{L}_{{1}}\Rightarrow{X}={A}+{t}{V}$$
$$\displaystyle{\left[{x},{y}\right]}={\left[-{3},{4}\right]}+{t}{\left[{1},{1}\right]}$$
$$\displaystyle{\left[{x},{y}\right]}={\left[-{3},{4}\right]}+{t}{\left[{t},{t}\right]}$$
$$\displaystyle{\left[{x},{y}\right]}={\left[-{3},+{t},{4}+{t}\right]}$$
in parametric form
$$\displaystyle{<}{x},{y}\ge{<}-{3}+{t},{4}+{t}{>}$$
Similarly,line $$\displaystyle{L}_{{2}}$$ also passes through (2,1)(A)
and direction vector [-3,1](v)
$$\displaystyle{L}_{{2}}={x}={a}+{t}{V}$$
$$\displaystyle{\left[{x},{y}\right]}={\left[{2},{1}\right]}+{t}_{{1}}{\left[-{3},{1}\right]}$$
$$\displaystyle{\left[{x},{y}\right]}={\left[{2}-{3}{t}_{{1}},{1}+{t}_{{1}}\right]}$$
$$\displaystyle{L}_{{2}}:\prec{x},{y}\ge{<}{2}-{3}{t}_{{1}},{1}+{t}_{{1}}{>}$$
Point of intersection of $$\displaystyle{L}_{{1}}{\quad\text{and}\quad}{L}_{{2}}$$ are:-
x=-4
So,Position vecor of their point of intersection
$$\displaystyle=-{4}{w}{i}{d}{e}\hat{{{i}}}+{3}{w}{i}{d}{e}\hat{{{j}}}$$

### Relevant Questions

write the general forms of the equations of the lines that pass through the point and are (a) parallel to the given line and (b) perpendicular to the given line.
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Write the general forms of the equations of the lines that pass through the point and are (a) parallel to the given line and (b) perpendicular to the given line. Point (−1, 0) Line y=-3
Write the general forms of the equations of the lines that pass through the point and are (a) parallel to the given line and (b) perpendicular to the given line. Point (−7, −2) Line x = 1
The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. Write the solution in vector form. $$\displaystyle{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}&−{1}&{3}&{9}\backslash{0}&{1}&{2}&−{5}&{8}\backslash{0}&{0}&{0}&{0}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$
1)What is the position vector r(t) as a function of angle $$\displaystyle\theta{\left({t}\right)}$$. For later remember that $$\displaystyle\theta{\left({t}\right)}$$ is itself a function of time.
Give your answer in terms of $$\displaystyle{R},\theta{\left({t}\right)}$$, and unit vectors x and y corresponding to the coordinate system in thefigure. 2)For uniform circular motion, find $$\displaystyle\theta{\left({t}\right)}$$ at an arbitrary time t.
Give your answer in terms of $$\displaystyle\omega$$ and t.
3)Find r, a position vector at time.
Give your answer in terms of R and unit vectors x and/or y.
4)Determine an expression for the positionvector of a particle that starts on the positive y axis at (i.e., at ,(x_{0},y_{0})=(0,R)) and subsequently moves with constant $$\displaystyle\omega$$.
Express your answer in terms of R, \omega ,t ,and unit vectors x and
Equations of lines Find both the parametric and the vector equations of the following lines.
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Equations of lines Find both the parametric and the vector equations of the following lines. The line through (0, 0, 1) in the direction of the vector $$v = ⟨4, 7, 0⟩$$
Find sets of parametric equations and symmetric equations of the line that passes through the given point and is parallel to the given vector or line. (For each line, write the direction numbers as integers.) Point $$(-1,\ 0,\ 8)$$ Parallel to $$v = 3i\ +\ 4j\ -\ 8k$$ The given point is $$(−1,\ 0,\ 8)\ \text{and the vector or line is}\ v = 3i\ +\ 4j\ −\ 8k.$$ (a) parametric equations (b) symmetric equations
Determine whether the given $$\displaystyle{\left({2}\ \times\ {3}\right)}$$ system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form
$$\displaystyle{x}={a}{t}\ +\ {b},\ {y}={c}{t}\ +\ {d},\ {z}={e}{t}\ +\ {f}.$$
$$\displaystyle{x}_{{{1}}}\ +\ {2}{x}_{{{2}}}\ -\ {x}_{{{3}}}={2}$$
$$\displaystyle{x}_{{{1}}}\ +\ {x}_{{{2}}}\ +\ {x}_{{{3}}}={3}$$