Vectors Algebra
Basic Concepts
* Quantities that are characterised only by their magnitude are called scalar quantities.
* Quantities that are characterised by their magnitude as well as direction are called vector quantities.
* A line segment with a given direction is called a directed line segment.
* A directed line segment AB is written as \overrightarrow {AB}
Magnitude of \overrightarrow {AB} = \overrightarrow {AB} or \overrightarrow {a}
\overrightarrow {(OP)} is the positional vector of P with respect to 0
∠a, ∠β and ∠γ are called the direction angles of\overrightarrow {p}
cos a ( ɭ ), cos β ( m ) and cos γ (n) are called the direction cosines of \overrightarrow {r}
ɭr ( a), mr (b ) and nr (c) are called the direction ratios of \overrightarrow {r}
* For any given vector, ɭ² + m² + n² = 1
* For any given vector, a² + b² + c² ≠ 1, unless r² or r = 1
Types of Vectors
* A vector whose initial and terminal points are the same is called a zero vector \overrightarrow {(o)}
* A vector whose magnitude is 1 is called a unit vector (â).
* Two or more vectors having the same initial point are called coinitial vectors.
* Two or more vectors that are parallel to the same line are called collinear vectors.
* Two vectors having the same direction and magnitude are called equal vectors.
* Two vectors with the same magnitude but direction are called negative vectors of each other.
* A vector which is independent of its position is called a free vector.
Addition of Vectors
* Triangle law of vector addition: The sum of two vectors representing two sides of a triangle taken in the same order is given by the vector representing the third side of the
triangle taken in the opposite order.
* Parallelogram law of vector addition: The sum of two vectors representing adjacent sides of a parallelogram is given by the diagonal of the parallelogram
passing through the meeting point of the two adjacent sides.
* Commutative Property: Given any two vectors \overrightarrow {a} and \overrightarrow {b,} + \overrightarrow {b} =
\overrightarrow {b} + \overrightarrow {a}
* Additive Identity: For any vector \overrightarrow {AB} , \overrightarrow {Ab} + \overrightarrow {0} =
\overrightarrow {0} + \overrightarrow {AB} = \overrightarrow {AB}
* Associative Property: Given any three vectors \overrightarrow {a} , \overrightarrow {b} and \overrightarrow {c}
\overrightarrow {(a} + \overrightarrow {b)} + \overrightarrow {c} = \overrightarrow {a} + \overrightarrow {(b} + \overrightarrow {c)}
Multiplication of a Vector by a Scalar
* If λ is a scalar and \overrightarrow {r} is any vector, then:
λ \overrightarrow {r}, is a vector whose magnitude is λ times vector \overrightarrow {r}, and direction depends on the sign of λ.
* A unit vector in the direction of \overrightarrow {r} is ř = \frac{1}{ \overrightarrow {r}} \overrightarrow {r} .
* If Î, Ĵ and ќ, are unit vectors along the X, Y and Z axes, respectively, and the coordination of point P are (x, y, z), then the position vector of point with
respect to 0 is x Î + y Ĵ + z ќ.
| \overrightarrow {r} | = \sqrt{x² + y² + z²}
Given two points, P_{1} ( x_{1} , y_{2} , z_{1}) and P_{2} ( x_{2} , y_{2} , z_{2}):
\overrightarrow {P^{1} P^{2}} = ( x_{2} - x_{1} )Î + ( y_{2} - y_{1} ) Ĵ + ( x_{2} - x_{1} )ќ
Section Formula
The position vector of point R that divides the line segment joining points A and B in the ratio m: n
* Internally is \overrightarrow {OR} =\frac{ \overrightarrow {mb + na}}{m + n}
* Externally is \overrightarrow {OR} =\frac{ \overrightarrow {mb - na}}{m - n}
* If m:n = 1:1, then \overrightarrow {OR} =\frac{ \overrightarrow {a + b}}{2}
Scalar Product
\overrightarrow {a} . \overrightarrow {b} = | \overrightarrow {a} | | \overrightarrow {b} | cos θ,
where 0° ≤ θ ≥ 180°
When θ = 0°:
\overrightarrow {a}. \overrightarrow {b}. = | \overrightarrow {a} | | \overrightarrow {b} |
When θ = 180° :
\overrightarrow {a}. \overrightarrow {b}. = – | \overrightarrow {a} | | \overrightarrow {b} |
If \overrightarrow {a}. \overrightarrow {b} = 0 ↔ \overrightarrow {a} ± \overrightarrow {b}
Properties of Scalar Product
Given vector \overrightarrow {a} = a_{1} Î +a_{2} Ĵ + a_{3} k and \overrightarrow {b} = b_{1} Î + b_{2} Ĵ + b_{3} k:
\overrightarrow {a}. \overrightarrow {b} = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}
Given three vectors, \overrightarrow {a} , \overrightarrow {b} and \overrightarrow {c} : \overrightarrow {a}
. (\overrightarrow {b} + \overrightarrow {c}) = \overrightarrow {a} . \overrightarrow {b} + \overrightarrow {a}. \overrightarrow {c}
Given two vectors, \overrightarrow {a}and \overrightarrow {b}, and a scalar,
λ : λ (\overrightarrow {a} . \overrightarrow {b}= ( λ \overrightarrow {a} ) . \overrightarrow {b} =
\overrightarrow {a}. ( λ \overrightarrow {b} )
The projection vector of vector \overrightarrow {a} on a directed line ɭ is given by:
\overrightarrow {b} = \overrightarrow {a} cos θ Where θ is the angle between \overrightarrow {a} and ɭ.
The projection of vector \overrightarrow {a} directed line ɭ is given by:
| \overrightarrow {p} | = | \overrightarrow {a} | cos θ Where θ is the angle between \overrightarrow {a} and ɭ.
The projection of vector \overrightarrow {a} along another vector \overrightarrow {b} is given by:
\overrightarrow {p}= \overrightarrow {a}. \overrightarrow {b} = \frac{1}{ | \overrightarrow {b} | } ( \overrightarrow {a}. \overrightarrow {b} )
Given a vector \overrightarrow {a} = a_{1} Î + a_{2} Ĵ + a_{3}k
a_{1} = projection of \overrightarrow {a} on the X – axis
a_{2} = projection of \overrightarrow {a} on the Y – axis
a_{3} = projection of \overrightarrow {a} on the Z – axis
The direction cosines o vector \overrightarrow {a} = a_{1} Î + a_{2} Ĵ + a_{3}k are given by:
cos α = \frac{ a_{1} }{ | \overrightarrow {a} | } cos β = \frac{ a_{2} }{ | \overrightarrow {a} | } cos γ = \frac{ a_{3} }{ | \overrightarrow {a} | }
Cross Product
In a right – handed coordinate system, if two vectors point along the positive and Y axes, then their cross product points along the positive Z − axis.
Given
\overrightarrow {p} ≠ \overrightarrow {o} and \overrightarrow {q} ≠ \overrightarrow {o} :
\overrightarrow {p} x \overrightarrow {q} = | \overrightarrow {p} | | \overrightarrow {q} | sin θ ň
Where θ is the angle between \overrightarrow {p} and \overrightarrow {q}such that 0° ≤ θ ≤ 180° .
And ň is a unit vector such that ň ± \overrightarrow {q}
If either \overrightarrow {p} or \overrightarrow {q} = \overrightarrow {o}:
\overrightarrow {p} x \overrightarrow {q} = \overrightarrow {o}
Properties of Cross Product
\overrightarrow {p} × \overrightarrow {q} = \overrightarrow {o} ↔ \overrightarrow {p} \| ,
Where \overrightarrow {p} ≠ \overrightarrow {o} and \overrightarrow {q} ≠ \overrightarrow {o}
If
\overrightarrow {p} ± \overrightarrow {q}, then = | \overrightarrow {p \times q} | = | \overrightarrow {p} | | \overrightarrow {q} |
If θ is the angle between two vectors
\overrightarrow {p} and \overrightarrow {q}, then sin θ = \frac{| \overrightarrow {p \times q} | }{ | \overrightarrow {p} || \overrightarrow {q} |}
\overrightarrow {p} x \overrightarrow {q} ≠ \overrightarrow {q} x \overrightarrow {p}
If \overrightarrow {p}and \overrightarrow {q}, represents adjacent sides of a triangle, then the area of the triangle
= \frac{1}{2} | \overrightarrow {p \times q} |
If \overrightarrow {p} and \overrightarrow {q} , represents adjacent sides of a parallelogram, then the area of the
parallelogram = | \overrightarrow {p \times q} |
For any two vectors \overrightarrow {a} = a_{1} Î + a_{2} Ĵ + a_{3}k and \overrightarrow {b} = b_{1} Î + b_{2} Ĵ + b_{3}k:
\overrightarrow {a} x \overrightarrow {b} = ( a_{2} b_{3} - a_{3} b_{2}) Î –
( a_{1} b_{3} - a_{3} b_{1}) Ĵ + ( a_{1} b_{2} - a_{2} b_{1})k = \begin{pmatrix}
\widehat{I} & \widehat{J} & k\\
a_{1} & a_{2} & a_{3}\\
b_{1} & b_{2} & b_{3}
\end{pmatrix}