A triad of unit vectors:
A physical quantity that can be described using both magnitude and direction is known as a vector.
If a physical quantity can be described only by its magnitude or size then it is known as a scalar. A Scalar doesn't have direction.
(a) Types of vectors: There are different types of vectors out of which Equal vectors; Unit vector and Zero vector are important.
- Equal Vectors: Two Vectors are said to be equal only if their magnitude as well as the direction are same, regardless of their initial points.
- Unit Vector: A vector whose magnitude is unity and has the direction as that of the given vector.
- Zero Vector: A zero vector has Zero magnitude but has no particular direction. This is consistent with the fact that it is orthogonal to every other vector. Alternatively, it points in every direction, but with zero magnitude, since if you take any vector and multiply it by zero, you get the zero vector.
Conversion of polar coordinates in to cartesian coordinates:
(b) Cartesian Coordinates: This coordinate system is also called rectangular coordinate system. A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. The coordinates are written as an ordered pair (x, y).
(c) Polar Coordinates: The polar coordinate system is a two – dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. The polar coordinate system specifies points by the distance from the center (called the radius r), and an angle from the horizontal (usually called theta, θ) i.e.,( r, θ). The distance from the pole is called the radial coordinate, and the angle is the angular coordinate, or the polar angle.
Cross Product of two vector a and b:
(d) Addition and Multiplication of vectors: If two vectors have the same direction, their
resultant has a magnitude equal to the sum of their magnitudes and will also have the same direction. Similarly, oriented
vectors can be subtracted in the same manner. Parallelogram law of addition of vectors is a method to add two vectors acting
simultaneously at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, then the
diagonal of the parallelogram passing through that point represents their resultant in magnitude and direction.
Multiplication of two vectors: There are two kinds of multiplication operations for vectors – dot product and cross product.
Dot Product of Vectors: It is the multiplication of one vector by the other vector to get a scalar. It is also
known as Scalar product.
Cross Product of vectors: It is the multiplication of one vector by a second vector so as to produce another
vector. It is called vector product of vectors.