A quantity that has magnitude as well as direction is called a vector. A directed line segment is a vector.
The point A from where the vector AB starts is called its initial point, and the point B where it ends is called its terminal point. The distance between initial and terminal points of a vector is called the magnitude (or length) of the vector. The arrow indicates the direction of the vector.
Position vector of a point P(x, y, z) is given as OP(=r) = xi + yj + zk, and its magnitude by √(x2 + y2 + z2). This form of any vector is called its component form. x, y and z are called as the scalar components.
The angles α, β, γ made by the vector r with the positive directions of x, y and z axes respectively, are called its direction angles. The cosine values of these angles (cos α, cos β and cos γ) are called direction cosines of the vector r, and usually denoted by l, m and n.
The numbers lr, mr and nr, proportional to the direction cosines are called as direction ratios of vector r, and denoted as a, b and c, respectively.
Type of Vectors
A vector whose initial and terminal points coincide, is called a zero vector (or null vector). Zero vector can not be assigned a definite direction as it has zero magnitude.
A vector whose magnitude is unity (1 unit) is called a unit vector.
Two or more vectors having the same initial point are called coinitial vectors.
Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.
Two vectors are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points
Negative of a Vector
A vector whose magnitude is the same as that of a given vector, but direction is opposite to that of it, is called negative of the given vector.
Addition of Vectors
If we have two vectors with the same direction, then the addition of them is equal to the sum of their magnitude and will has the same direction.
If we have two vectors with different direction, then vector addition can take place by triangle law or parallelogram law.
Triangle Law of Vector Addition
If we have two vectors, then to add them, they are positioned so that the initial point of one coincides with the terminal point of the other.
Parallelogram Law of Vector Addition
If we have two vectors represented by the two adjacent sides of a parallelogram in magnitude and direction, then their sum is represented in magnitude and direction by the diagonal of the parallelogram through their common point.
a + b = b + a [Commutative Property]
(a + b) + c = a + (b + c) [Associative Property]
Multiplication of Vector by Scalar
The multiplication of a given vector by a scalar λ, changes the magnitude of the vector by the multiple |λ|, and keeps the direction same (or makes it opposite) according as the value of λ is positive (or negative).
Product of Two Vectors
Multiplication of two vectors is defined in two ways:
- Scalar (or dot) product where the result is a scalar
- Vector (or cross) product where the result is a vector
Scalar (or dot) Product
The scalar product of two non-zero vectors a and b is
a . b = ab cos θ
The dot product of two vectors is zero if and only if they are perpendicular to each other.
Vector (or cross) Product
The vector product of two non-zero vectors a and b is
a x b = ab sin θ n
The cross product of two vectors is zero if and only if they are parallel (or collinear) to each other.