Subtraction of vectors
While doing the subtraction of vectors, we change the direction of the vector to be subtracted and then add.
𝐴→−𝐵→=𝐴→+(−𝐵→)
Null vector
If vector A is multiplied by zero, we get a vector whose magnitude is zero, called a null vector or zero vector. The unit of the vector does not change on being multiplied by a dimensionless scalar.
Properties of null vector
1) It has an arbitrary direction.
2) It is represented by a point.
3) It has zero magnitude.
4) Dot product of a null vector with any vector is always zero.
5) Cross product of a null vector with any vector is also a null vector.
6) When a null vector is added or subtracted from a given vector, the resultant vector is the same as the given vector.
Orthogonal unit vectors
The unit vectors along the X-axis, Y-axis, and Z-axis of the right-handed cartesian coordinate system are written as
𝑖^,𝑗^ and 𝑘^
, respectively. They are known as orthogonal unit vectors.
Unit Vector
A vector having a unit magnitude is called a unit vector. It is used to denote the direction of a given vector.
𝐴→=𝑎^𝐴
𝑎^ is the unit vector along the direction of 𝐴→
.
Types of Vectors
(i) Negative of a vector: It has the same magnitude but opposite direction of the given vector.
(ii) Equal vectors: If two vectors have equal magnitude and direction, they are equal vectors.
(iii) Collinear vectors: Two vectors acting along the same straight lines or along parallel straight lines in the same direction or in the opposite direction are called collinear vectors.
(iv) Coplanar vectors: If three or more vectors lie in the same plane, then they are called coplanar vectors.
(v) Zero vector: It is a vector with zero magnitude and no specific direction.
Addition of Vectors
Law of triangle: If two sides of a triangle are shown by two continuous vectors (vector A and vector B), then the third side of the triangle in the opposite direction shows the resultant of two vectors (vector C).
𝐶→=𝐴→+𝐵→
Vector addition is commutative.
⇒ A + B = B + A
Vector addition is associative.
⇒ A+(B+C) = (A+B)+C
If all sides of a polygon are represented by continuous vectors, the vector sum of all sides is zero.
Polygon method: We use this method when we have to add more than two vectors. It is an extension of the triangular law of vector addition. If a number of vectors can be represented in magnitude and direction by the sides of a polygon taken in the same order, then their resultant is represented in magnitude and direction by the closing side of the polygon taken in the opposite order.
Representation of a Vector
A vector is represented by a line with an arrowhead. The point O from which the arrow starts is called the tail or initial point, or origin of the vector. Point A, where the arrow ends, is called the tip or head or terminal point of the vector. A vector displaced parallel to itself remains unchanged. If a vector is rotated through an angle other than 3600, it changes.
A vector can be replaced by another when its direction and magnitude are the same
#What Is a Scalar Quantity?
A scalar quantity is defined as the physical quantity with only magnitude and no direction. Such physical quantities can be described just by their numerical value without directions. The addition of these physical quantities follows the simple rules of algebra, and here, only their magnitudes are added.
Examples of Scalar Quantities
Some examples of scalar include:
Mass
Speed
Distance
Time
Volume
Density
Temperature
What Is a Vector Quantity?
A vector quantity is defined as the physical quantity that has both directions as well as magnitude.
A vector with a value of magnitude equal to one is called a unit vector and is represented by a lowercase alphabet with
Examples of Vector Quantities
Examples of vector quantity include:
Linear momentum
Acceleration8
Displacement
Momentum
Angular velocity
Force
Electric field
Polarization
Difference Between Scalars and Vectors
The difference between Scalars and Vectors is crucial to understand in physics learning.
Components of a vector
Consider a vector, V. The components of a vector in a 2D coordinate system are considered to be x-component and y-component. We can represent V = (vx, vy). Let θ be the angle formed between the vector V and the x-component of the vector. The vector V and its x-component (vx) form a right-angled triangle if we draw a line parallel to the y-component (vy).
The horizontal component vx = V cos θ.
The vertical component vy = V sin θ.
Position Vectors
A position vector is a vector that gives the position of a point with respect to the origin of the coordinate system. The magnitude of the position vector is the distance of the point P from the origin O. Vector OP is the position vector that gives the position of the particle with reference to O.
Consider point P, whose coordinates are (x, y).
𝑂𝑃→=𝑟→
x = r cos θ, y = r sin θ
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