In our day-to-day life, we come across many quantities that have only magnitude. Such as length, temperature, time, speed, area, volume etc. Such quantities are called Scalars. On the other hand, some quantities that involve magnitude and direction are called Vectors. For example, force, weight, momentum, acceleration, force, etc. It is also known as the Euclidean vector or just a “vector “.
Coordinate plane
The coordinate plane dates back hundreds of years. Cartesian coordinates were invented by René Descartes, a French mathematician. A coordinate plane is a two-dimensional plane produced by the intersection of the y-axis and the x-axis. The origin is the place where two perpendicular lines intersect at zero. The axes divide the coordinate plane into four equal portions, each of which is referred to as a quadrant.
Definition of Vector
An object that has both magnitude and direction is referred to as a vector. A vector’s starting point is called ‘Tail’, and its finishing point (which has an arrow) is called ‘Head’.
The direction of the vector is from the tail to the head. The length of the straight line determines the magnitude of the vector, while the arrowhead determines the direction. Two vectors are deemed same if their magnitude and direction are the same. The magnitude of a vector does not change whether it is rotated or moved around.
Vector Representation
A vector has both magnitude and direction. The magnitude is the length of the line AB, and the arrow’s head points in the direction. As a result, vectors between two points A and B are denoted by the letters ▁AB or vector a. The vector’s direction is shown by the arrow above its head.
The starting point A of the vector ▁AB is termed its initial point, and the ending point B is called its terminal point. The length (magnitude) of a vector is the distance between its beginning and terminal points, represented as I▁AB I or I▁a I, or a. The vector’s direction is shown by the arrow.〖^〗
The Magnitude of a Vector
A vector’s magnitude is indicated by vertical lines on both sides of it . It represents the length of the vector. By using the Pythagoras theorem the magnitude of a vector may be computed mathematically, is used.
|a|=√ x² + y²
Angle Between Two Vectors
The shortest angle at which any of two vectors can be rotated around the other vector so that both vectors have the same direction is called an angle between two vectors.
If two vectors, let’s call them a and b, are on a plane and their tails are joined, they form an angle θ. Now the value θ can be calculated using the below formula
a.b = |a| |b|.cosθ
Types of Vectors
- The zero vector has a magnitude of zero and its initial and terminal points coincide.
- A unit vector is defined as a vector with a magnitude of one unit.
- A position vector represents the position of a point in reference to its origin.
- Co initial vectors are two or more vectors that have the same starting point.
- Coplanar vectors are vectors that are in the same plane.
- Collinear vectors are two vectors that lie on the same straight line irrespective of tgeir magnitudes and direction
- A negative vector is another vector of same magnitude but moving in the opposite direction.
Examples
Example1: Find the magnitude of vector c = (3, 4)
Solution: To calculate the magnitude of vector c, we do the following:
- |c|=√ x2 + y2
- |c| = √ (3)2 + (4)2
- |c| = √ 9 + 16
- |c| = √ 25
- ∴ |c| = 5
Example 2: Suppose there are two vectors and , if |a| = 7 units and |b| = 12 units, find the dot product if the angle of separation between vector ‘a’ and ‘b’ is 60.
Solution: The dot product of two vectors can be calculated as follows:
- a⋅b=|a||b|cosθ
- a⋅b=|a||b|cosθ
- a.b =7*12* cos 60
- a.b=84 * 1/2
- ∴ a.b=42
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