Direction Cosines and Direction Ratios
* Directional angles: The angles made by a line with the positive directions of the X, Y and Z axes are called directional angles.
* Directional Cosines: If α, β and γ are the directional angles of a directed line L, them cos α, cos β, and cos γ , are called the directional cosines of the directed
line L.
* If ɭ, m, n are the directional cosines of a line, then ɭ² + m² +n² = 1
* Direction ratios: Any three numbers that are proportional to the direction cosines of a line are called the direction ratios of the line.
If a, b, c are a set of the directional ratios of a line, then its direction cosines are:
ɭ = ± \frac{a}{ \sqrt{a² + b² + c²} } m = ± \frac{b}{ \sqrt{a² + b² + c²} } n = ± \frac{c}{ \sqrt{a² + b² + c²} }
Directional Cosines and Direction Ratios
* Direction cosines of a line passing through two points
( x1, y1, z1 ) and ( x2, y2, z2 )
cos a ( ɭ )= \frac{ x_{2} - x_{1} } { \sqrt{( x_{2} - x_{1})² + ( y_{2} - y_{1})² + ( z_{2} } - z_{1} )² }
cos a ( β) = \frac{ y_{2} - y_{1} } { \sqrt{( x_{2} - x_{1})² + ( y_{2} - y_{1})² + ( z_{2} } - z_{1} )² }
cos a ( γ ) = \frac{ z_{2} - z_{1} } { \sqrt{( x_{2} - x_{1})² + ( y_{2} - y_{1})² + ( z_{2} } - z_{1} )² }
Direction ratios of a line passing through (x_{1} , y_{1} , z_{1}) and (x_{2} , y_{2} , z_{2}):
(x_{2} - x_{1}), (y_{2} - y_{1}) (z_{2} - z_{1}) or (x_{1} - x_{2}), (y_{1} - y_{2}), (z_{1} - z_{2})
Equation of Line
* The vector equation of a line that passes through a given point with position vector \overrightarrow {a} and is parallel to vector \overrightarrow {b} is given by: \overrightarrow {r} = \overrightarrow {a} + λ \overrightarrow {b}
* The Cartesian equation of a line with direction ratios a, b, c that passes through a point (x_{1} , y_{1} , z_{1}) is given by:
\frac{x - x_{1} }{a} = \frac{y - y_{1} }{b} = \frac{z - z_{1} }{c}
* The Cartesian equation of a line with direction cosines l, m, n that passes through a point (x_{1} , y_{1} , z_{1}) is given by:
\frac{x - x_{1} }{l} = \frac{y - y_{1} }{m} = \frac{z - z_{1} }{n}
* The vector equation of a line that passes through two given points with position vector \overrightarrow {a} and \overrightarrow {b} is given by:
\overrightarrow {r} = \overrightarrow {a} + λ ( \overrightarrow {b} – \overrightarrow {a} )
* The Cartesian equation of a line that passes through two given points (x_{1} , y_{1} , z_{1}) and (x_{2} , y_{2} , z_{2}) is given by:
\frac{x - x_{1} }{ x_{2} - x_{1} } = \frac{y - y_{1} }{ y_{2} - y_{1} } = \frac{z - z_{1} }{ z_{2} - z_{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 = | \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_{1} - 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 = | l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} |
sin θ = \sqrt{( l_{1} m_{2} - l_{2} m_{1})² + ( m_{1} n_{2} - m_{2} n_{1} )² + ( n_{1} l_{2} - n_{2} l_{1} )² }
If θ is the angle between two lines with direction cosines \overrightarrow {r} = \overrightarrow {a_{1}} + \overrightarrow {λb_{1}} and \overrightarrow {μb_{2}}
cos θ = | \frac{\overrightarrow { b_{1}}. \overrightarrow { b_{2} }}{ | \overrightarrow { b_{1} }. | \overrightarrow { b_{2} } | } |
If θ is the 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_{2} }{ c_{2} }
cos θ = | \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} a_{2} ) 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} }
Shortest Distance between Two Lines
* A pair of coplanar lines can either be parallel lines or intersecting lines.
* Two lines are neither parallel nor intersecting are called skew lines. skew lines are always non – coplanar.
* The shortest distance between two skew lines is the length of the shortest line segment that joins a point on one line to a point on the other line.
The line segment is perpendicular to both the lines.
* The shortest distance d between two skew lines \overrightarrow {r} = \overrightarrow {a_{1}} +\overrightarrow {λb_{1}} and \overrightarrow {r} = \overrightarrow {a_{2}} +\overrightarrow {μb_{2}}
is given by:
d = | \frac{\overrightarrow {( b_{1} X} \overrightarrow { b_{2}) }}{ \overrightarrow { b_{1} X} \overrightarrow { b_{2} }} |
* The shortest distance d between two skew lines in Cartesian form
\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_{2} }{ c_{2} } is given by:
d = | \begin{pmatrix}
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{pmatrix} |
\sqrt{( b_{1} c_{2} - b_{2} c_{1} )² + ( c_{1} a_{2}+ c_{2} a_{1} )² + ( a_{1} b_{2} + a_{2} b_{1} )² }
* The shortest distance d between two parallel lines \overrightarrow {r} = \overrightarrow {a_{1}} +\overrightarrow {λb} and \overrightarrow {r} = \overrightarrow {a_{2}} +\overrightarrow {μb} is given by:
d = | \frac{\overrightarrow {b}X ( \overrightarrow { a_{2}- a_{1} }) }{ | \overrightarrow {b} | } |
Equation of Planes – I
The vector equation of plane in normal form is given by \overrightarrow {r} . ň = d, Where:
\overrightarrow {r} = Position vector of a point on the plane.
ň = Unit vector along the normal to the plane.
d = Distance of the plane from the origin
* The Cartesian equation of a plane in normal form is given by
lx + my + nz = d, where:
l, m, n are Direction cosines of unit vector ň along the normal to the plan
d = Distance of the plan from the origin
* The vector equation of plan passing through a given point A with position vector \overrightarrow {a} and perpendicular to a given vector \overrightarrow {N} is: ( \overrightarrow {r} − \overrightarrow {a} ) . \overrightarrow {N} = 0
* The Cartesian equation of a plane passing through a given point A and perpendicular to a given vector \overrightarrow {N} is lx + my + nz = d,
where: A, B, C are Direction ratios of vector \overrightarrow {N}
(x_{1 } , y_{1} , z_{1}) = Coordinates of point A
Equation of Planes – II
* The vector form of the equation of a plane passing through three non – collinear points having position vectors \overrightarrow {a} \overrightarrow {b} and \overrightarrow {c} is: ( \overrightarrow {r} − \overrightarrow {a} ) . [ ( \overrightarrow {b} − \overrightarrow {a} ) X ( \overrightarrow {c} − \overrightarrow {a} ) ]
* The Cartesian form of equation of a plane passing through three non – collinear points having the coordinates
(x_{1 } , y_{1} , z_{1}) , (x_{2 } , y_{2} , z_{2}) and (x_{3 } , y_{3} , z_{3}) is:
| \begin{pmatrix}
x - x_{1} & y- y_{1} & z - z_{1} \\
x_{2} - x_{1} & y_{2}- y_{1} & z_{2}- z_{1} \\
x_{3} - x_{1} & y_{3} - y_{1} & z_{3}- z_{1}
\end{pmatrix} | = 0
* The equation of a plane making intercepts a , b and c, on the x , y and z axes,
respectively, is: \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1
* The vector form of the equation of a plane passing through the intersection of two given planes \overrightarrow {r}. ň = d2 and \overrightarrow {r} . ň2 d2 is: \overrightarrow {r} . = ( \overrightarrow { n{1}}+ \lambda \overrightarrow { n{2}}) = d1 + λd2
* The Cartesian form of equation of a plane passing through the intersection of two given planes is:
( a_{1} x + b_{1} y + c_{1}z - d_{1} ) + ( a_{2} x + b_{2} y + c_{2}z - d_{2} ) = 0
Coplanarity of Lines and Angle between Planes
* Two lines given by \overrightarrow {r} = \overrightarrow {a_{1}} + \overrightarrow {λb_{1}} and \overrightarrow {r} = \overrightarrow {a_{2}} + \overrightarrow {μb_{2}}are coplanar if and only if:
( \overrightarrow {a_{2}} − \overrightarrow {a_{1}} ). ( \overrightarrow {b_{1}} × \overrightarrow {b_{2}} ) = 0
* Two lines l_{1} and l_{2} are coplanar if:
\begin{pmatrix}
x_{2} - x_{1} & y_{2} - y_{1} & z_{2} - z_{1} \\
a_{1} & b_{1} & c_{1}\\
a_{2} & b_{1} & c_{2}
\end{pmatrix}
Where:
* ( x_{1 } , y_{1} , z_{1}) are the coordinates of a point on line l_{1} , and a_{1 } , b_{1} , c_{1} are the direction ratios of a vector parallel to line l_{1}
* ( x_{2 } , y_{2} , z_{2}) are the coordinates of a point on line l_{1} , and ( a_{2 } , b_{2} , c_{2}) are the direction ratios of a vector parallel to line l_{2}
* Angle θ between two planes \overrightarrow {r}. \overrightarrow {n_{1}} = b_{1} is given by:
cos θ = | \frac{\overrightarrow { n_{1}}. \overrightarrow { n_{2} }}{ | \overrightarrow { n_{1} }. | \overrightarrow { n_{1} } | } |
* Angle θ between two planes a_{1}x + b_{1}y + c_{1}z – d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z – d_{2}) = 0
cos θ = | \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 given planes are perpendicular if:
\overrightarrow { n_{1} }. \overrightarrow { n_{2} } = 0 or a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0
* Two given planes are parallel if: \overrightarrow { n_{1} }. \overrightarrow { n_{2} } = | \overrightarrow { n_{1} } | | \overrightarrow { n_{2} } | or \frac{ a_{1} }{ a_{2} } = \frac{ b_{1} }{ b_{2} } = \frac{ c_{1} }{ c_{2} }
Distance of a Point from a Plane, and Angle between a Line and a Plane
* The perpendicular distance of a point with position vector \overrightarrow {a} from a plane \overrightarrow {r}.
ň = d is given by:
| d - \overrightarrow {a}. \check{n} |.
* The perpendicular distance of a point with position vector \overrightarrow {a} from a plane \overrightarrow {r} . \overrightarrow {N} = d is given by:
| \frac{\overrightarrow {a}. \overrightarrow {N}-}{ | \overrightarrow {N} | } d
* The perpendicular distance of a plane \overrightarrow {r}. \overrightarrow {N} = d from the origin is given by
\frac{ | d | }{ | \overrightarrow {N} | }
* The perpendicular distance of a point A ( x_{1} + y_{1} + z_{1} ) from a plane Ax + By + Cz
= D is given by
\frac{ | A x_{1} + B y_{1} + C z_{1}- D | }{ | \sqrt{A² + B² + C² | } }
* Angle ф between a line \overrightarrow {r} = \overrightarrow {a} + λ\overrightarrow {b} and a plane
\overrightarrow {r}. \overrightarrow {n} = d is given by
ф = sin^{-1} \frac{ |\overrightarrow {b}. \overrightarrow {n} | }{ | \overrightarrow {b} | | \overrightarrow {n} | }