In our daily life, we generally come across two types of quantities, namely scalars and vectors.
Scalars: a quantity that has magnitude only is known as a scalar.
Examples: Each of the quantities mass, length, time etc are scalars.
Vectors: A quantity that has magnitude as well as direction is known as a vector.
Examples: each of these quantities force, velocity, acceleration is a vector.
We define vector as “a directed line segment”
A directed line segment with an initial point A and a terminal point B is a vector donated by .
The magnitude of is denoted by | |.
Some commonly used terminology is given below
Unit vector: A vector .
Equal vectors: Two vectors and are said to be equal if they have the same magnitude and the same direction regardless of the positions of their initial points.
Negative of a vector : A vector having the same magnitude as that of a given vector and a direction opposite to that of the is known as the negative of the given vector , to be denoted by .
Thus if . = , then = –
Zero or null vector: A vector whose initial and terminal points coincide is known as a zero vector, denoted by .
Cleary the magnitude of a zero vector is zero but it cannot be assigned any definite direction.
Coinitial vectors: two or more vectors having the same initial points are known as coinitial vectors.
Collinear vectors: two vectors having the same or parallel supports are known as collinear vectors.
Like Vector: Collinear vectors having the same direction are known as like vectors.
Unlike Vectors: Collinear vectors having opposite directions are known as unlike vectors.
Free Vectors: If the initial point of a vector is not specified it is known as a free vector.
Localized Vector: I vector drawn parallel to a given vector through a specified point as the initial point is known as localized vector.
Coplanar Vectors: three or more non zero vectors lying in the same plane or parallel to the same plane are said to be coplanar, otherwise they are known as non-coplanar.
Position of a vector: let O be the origin of and let A be a point such that = , then we say that the position vector of A is .
Algebraic Properties of Vectors
- Commutative (vector) P + Q = Q + P
- Associative (vector) (P + Q) + R = P + (Q + R)
- Additive identity There is a vector 0 such
that (P + 0) = P = (0 + P)
for all P
- Additive inverse For any P there is a vector -P such that P + (-P) = 0
- Distributive (vector) r(P + Q) = rP + rQ
- Multiplicative identity: For the real number 1, 1.P = P for each P
Scalar product of two vectors
Let a and b be two vectors and let ? be the angle between them. Then, the scalar product or dot product of a and b is defined as
a.b =|a| |b| cos ? = abcos?
Vector product of two vectors
Let a and b be two nonzero, nonparallel vectors, and let ? be the angle between them such that 0< ?<?.
Then, the vector product of a and b is defined as
a * b= (|a| |b| sin ?)