Straight Lines

General equation of the line

The general equation of a straight line in the cartesian coordinate system is given by \(A x+B y+C=0\), where the vector \((A, B)\) is the normal vector to the straight line.

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Slope-Intercept form of the line

The slope-intercept form of the line is given by

\[y=m x+c\]

where \(m\) is the slope and \(c\) is the \(y-intercept\).

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Equation of a line passing through two points

The equation of a straight line passing through two-points \((x_1, y_1)\) and \((x_2, y_2)\) is given by

\(\dfrac{y-y_{1}}{y_{2}-y_{1}}=\dfrac{x-x_{1}}{x_{2}-x_{1}}\) or \(\begin{vmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{vmatrix}=0\)

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Intercept form of the Line

The intercept form of a straight line is given by

\[\dfrac{x}{a}+\dfrac{y}{b}=1\]

where \(a\) and \(b\) are \(x-intercept\) and \(y-intercept\) respectively.

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Vector equation of the Line

The vector equation of the line is given by

\[\vec{r} = \vec{a}+t\vec{b}\]

where \(\vec{a}\) lies on the line and \(\vec{b}\) determines the direction of the line.

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Point direction form of the Line

The point direction form of the line is given by

\[\dfrac{x-x_{1}}{a}=\dfrac{y-y_{1}}{b}=\dfrac{z-z_{1}}{c}\]

where \(P_{1}\left(x_{1}, y_{1}, z_{1}\right)\) lies on the line and \(s=(a,b,c)\) is the direction vector of the line.

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Distance from a Point to a Line

The distance \(d\) from a point \(M(x_1,y_1)\) to the line \(A x+B y+C=0\) is given by:

\[d=\dfrac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\]

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Condition for two lines to be Parallel

• Two straight lines \(y=m_{1} x+c_{1}\) and \(y=m_{2} x+c_{2}\) are parallel if \(k_1=k_2\).

• Similarly, \(A_{1} x+B_{1} y+C_{1}=0\) and \(A_{2} x+B_{2} y+C_{2}=0\) are parallel if \(\dfrac{A_{1}}{A_{2}}=\dfrac{B_{1}}{B_{2}}\).

• Also, if the lines are represented in point direction forms, then they are parallel if their direction vectors \(\mathbf{s}_{\mathbf{1}}\left(a_{1}, b_{1}, c_{1}\right)\) and \(\mathbf{s}_{2}\left(a_{2}, b_{2}, c_{2}\right)\) are collinear, i.e., \(\dfrac{a_{1}}{a_{2}}=\dfrac{b_{1}}{b_{2}}=\dfrac{c_{1}}{c_{2}}\).

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Condition for two lines to be Perpendicular

• Two straight lines \(y=m_{1} x+c_{1}\) and \(y=m_{2} x+c_{2}\) are perpendicular if \(k_{1} k_{2}=-1\). Similarly, \(A_{1} x+B_{1} y+C_{1}=0\) and \(A_{2} x+B_{2} y+C_{2}=0\) are perpendicular if \(A_{1} A_{2}+B_{1} B_{2}=0\).

• Also, if the lines are represented in point direction forms, then they are perpendicular if the dot product of their direction vectors is 0, i.e., \(s_1 \cdot s_2=0\), or \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\).

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Angle between two lines

• The angle between two lines is given by:

\[\tan \varphi=\dfrac{m_{2}-m_{1}}{1+m_{1} m_{2}}\]

when lines are represented in slope-intercept forms,

\[\cos \varphi=\dfrac{A_{1} A_{2}+B_{1} B_{2}}{\sqrt{A_{1}^{2}+B_{1}^{2}} \sqrt{A_{2}^{2}+B_{2}^{2}}}\]

when lines are represented by their general equations, and \(\cos \varphi=\dfrac{\mathbf{s}_{1} \cdot \mathbf{s}_{2}}{\left|\mathbf{s}_{1}\right| \cdot\left|\mathbf{s}_{2}\right|}=\dfrac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \cdot \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\)

when the lines are represented in their point direction forms.

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Condition of Interesection of two lines

• Two lines \(\dfrac{x-x_{1}}{a_{1}}=\dfrac{y-y_{1}}{b_{1}}=\dfrac{z-z_{1}}{c_{1}}\) and \(\dfrac{x-x_{2}}{a_{2}}=\dfrac{y-y_{2}}{b_{2}}=\dfrac{z-z_{2}}{c_{2}}\) intersect if:

\[\begin{vmatrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix}\]

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Intersection point of two lines

• The intersection point of two straight lines \(A_1 x+B_1 y+C_1=0\) and \(A_2 x+B_2 y+C_2=0\) is given by:

\[x_0=\dfrac{-C_1 B_2+C_2 B_1}{A_1 B_2-A_2 B_1}\] \[y_0=\dfrac{-A_1 C_2+A_2 C_1}{A_1 B_2-A_2 B_1}\]

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• The straight line given by \(\dfrac{x-x_{1}}{a}=\dfrac{y-y_{1}}{b}=\dfrac{z-z_{1}}{c}\) and the plane given by \(A x+B y+C z+D=0\) are:
  (i) Parallel, if \(\mathbf{n} \cdot \mathbf{s}=0\), or \(A a+B b+C c=0\)
  (ii) Perpendicular, if \(\mathbf{n} \| \mathbf{s}\) or \(\dfrac{A}{a}=\dfrac{B}{b}=\dfrac{C}{c}\)

Shortest Distance Between Two Skew Lines

The shortest distance between two skew lines is the length of the mutual perpendicular between the two lines.

Let the two lines be given by: \(\vec{r}_{1} =\vec{a}_{1}+t \cdot \vec{b}_{1}\)

\[\vec{r}_{2} =\vec{a}_{2}+t \cdot \vec{b}_{2}\]

where \(\mathrm{P}=\vec{a}_{1}\) lies on \(l_1\) and \(\mathrm{Q}=\vec{a}_{2}\) lies on \(l_2\).

The unit vector normal to both \(l_1\) and \(l_2\) is given by \(\dfrac{\left(\vec{b}_{1} \times \vec{b}_{2}\right)}{\vert \vec{b}_{1} \times \vec{b}_{2} \vert}\).

Now, the projection of \(PQ\) on the normal will be the shortest distance between \(l_1\) and \(l_2\), and is given by:

\[\dfrac{\vert\left(\vec{a}_{2}-\vec{a}_{1}\right) \cdot \left(\vec{b}_{1} \times \vec{b}_{2}\right)\vert}{\vert\vec{b}_{1} \times \vec{b}_{2}\vert }\]

Straight Lines

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