Slope of a Line
* The angle made by a straight line in the anti-clockwise direction with the x-axis is called inclination.
Slope of a line (m) = tan θ
Where θ = inclination of the line
If θ = 0o
Slope (m) = 0
If θ = 900
Slope (m) is not defined.
* The slope of a line passing through points ( x_{1} , y_{1} ) and ( x_{2} , y_{2} ) is \frac{ y_{2} - y_{1} }{ x_{2} - x_{1} }
Angle Between two Lines
* If θ is the angle between two lines with direction ratios a_{1} , b_{1} , c_{1} and a_{2} , b_{2} , c_{2} :
Cos θ = \Large | \frac{ a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} }{ \sqrt{ a_1^2 + b_1^2 + c_1^2 } \sqrt{ a_2^2 + b_2^2 + c_2^2 } } |
Sin θ = \sqrt{ \frac{( a_{1} b_{2} - a_{2} b_{1} )² + ( b_{1} c_{2} - b_{2} c_{1} )² + ( c_{1} a_{2}- c_{2} a_{1} )² }{( a_1^2 + b_1^2 + c_1^2 ) ( a_2^2 + b_2^2 + c_2^2 ) } }
* If θ is the angle between two lines with direction cosines l_{1} , m_{1} , n_{1} and l_{2} , m_{2} , n_{2} :
Cos θ =|l1 l2 + m1 m2 + n1 n2|
Sin θ = \sqrt{( l_{1} m_{2} + l_{2} m_{2})2 ( m_{1} n_{2} − m_{2} n_{1} )2 + ( n_{1} n_{2} − n_{2} l_{1} )2}
If θ is the angle between two lines \overrightarrow {r} = \overrightarrow { a_{1} } \overrightarrow { λb_{1} } and \overrightarrow {r} = \overrightarrow { a_{2} } + \overrightarrow { μb_{2} }:
Cos θ = | \frac{ \overrightarrow { b_{1}}. \overrightarrow { b_{2}}}{ | \overrightarrow { b_{1}} | \overrightarrow { b_{2}} | } |
If θ is angle between two lines : \frac{x - x_{1} }{ a_{1} } = \frac{y - y_{1} }{ b_{1} } = \frac{z - z_{1} }{ c_{1} } and \frac{x - x_{2} }{ a_{2} } = \frac{y - y_{2} }{ b_{2} } = \frac{z - z_{1} }{ c_{2} }
Cos Θ = \Large | \frac{ a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} }{ \sqrt{ a_1^2 + b_1^2 + c_1^2 } \sqrt{ a_2^2 + b_2^2 + c_2^2 } } |
* Two lines with direction ratios a_{1} , b_{1} , c_{1} and a_{2} , b_{2} , c_{2} are perpendicular if:
a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0
Two lines with direction ratios a_{1} , b_{1} , c_{1} and a_{2} , b_{2} , c_{2} are parallel if:
\frac{ a_{1} }{ a_{2} } = \frac{ b_{1} }{ b_{2} } = \frac{ c_{1} }{ c_{2} } = 0
Equations of Lines (Part 1)
* A linear equation in two variables represents the equation of a line.
* The condition for a point to lie on a line is that the point has to satisfy the equation of the line.
* The point of intersection of two lines is obtained by solving the equations of the two lines.
Equation of horizontal line:
y = a or y = – a
Equation of a vertical line:
x = a or x = – a
Point-Slope Form:
y – y_{1} = m(x – x_{1} )
Two-Point Form:
y – y_{1} = ( \frac{ y_{2} - y_{1} } { x_{2} - x_{1} } ) × (x – x_{1} )
Equations of Lines (Part 2)
* The distance from the origin at which a lines cuts an axis is called the intercept of the line on the on that axis.
Slope-Intercept Form (Using y-intercept): y = mx + c
Slope-Intercept Form (Using x-intercept): y = m(x – d)
Intercept Form: \frac{ x } { a} + \frac{ y } { b} = 1
Normal Form: x cos ω + y sin ω = p
General Equation of a Line
* General Equation of a Line: Ax + By + c = 0,
Where A and B ≠ 0 simultaneously.
General Equation in Slope-Intercept Form (y = mx + c) is:
Y = ( \frac{ - A } { B} ) x + (- \frac{ - C } { B} ), where: B ≠ 0
Slope (m) = \frac{ - A } { B}
Y-intercept (c) = – \frac{ C } { B}
General Equation in Intercept Form ( \frac{ x } { a} + \frac{y}{b} = 1 ) is:
\frac{ x } { ( \frac{- C}{A} )} + \frac{y}{ \frac{- C}{B} } = 1, where: c ≠ 0
x-intercept (a) = \frac{- C}{A}
y-intercept (b) = \frac{- C}{B}
General Equation in Normal Form (x cos ω) + y sin ω) = p) is:
x ( \pm \frac{A}{ \sqrt{( A² + B²)} } ) + y ( \pm \frac{B}{ \sqrt{( A² + B²)} } ) = \pm \frac{C}{ \sqrt{( A² + B²)} }, where:
cos ω = \pm \frac{A}{ \sqrt{( A² + B²)} },
sin ω = \pm \frac{B}{ \sqrt{( A² + B²)} },
p = \pm \frac{C}{ \sqrt{( A² + B²)} }
Distance Between Parallel Lines
Distance (d) = |\frac{ Ax_{1} + By_{1} + C}{ \sqrt{A²+B²} } |
Distance (d) between parallel lines y = mx + c_{1}
Y = mx + c_{2} is given by: d = | \frac{C_{1} + C_{2} }{ \sqrt{1 + m²} } |
Distance (d) between parallel lines Ax + By + C1 and
Ax + By + C_{2} is given by: d = | \frac{C_{1} + C_{2} }{ \sqrt{A²+ B²} } |