Table of contents | |
Scalars | |
Vectors | |
General Points Regarding Vectors | |
Scalar or Dot Product of Two Vectors | |
Vector or Cross Product of Two Vectors |
Different physical quantities can be classified into the following two categories:
Scalar quantity is defined as the physical quantity that has magnitude but no direction.
A vector quantity is defined as the physical quantity that has both direction as well as magnitude and follows law of vector addition.
Vectors
For example, Speed = 4 m/s (is a scalar), Velocity = 4 m/s toward north (is a vector).
Difference between scalar and vector quantities is mentioned in table below:
Vectors are classified into two types such as :
Vectors which have a starting point or a point of application are called polar vectors.
Examples are Force, Displacement, etc.
Polar Vectors
Vectors that represent effects of rotation and are directed along the axis of rotation.
Examples of axial vectors include angular velocity, angular momentum, torque, etc.
Axial Vectors
The direction of an axial vector is determined by the direction of rotation:
The angle between two vectors refers to the smaller angle formed between them when they are positioned tail to tail by moving one of the vectors parallel to itself.
A vector having magnitude equal to unity but having a specified direction is called a unit vector.
A unit vector of A is written as (A cap).
It is expressed as
Q1.
Sol:
A null vector is a vector that has zero magnitude but an undefined or arbitrary direction.
It is also called a zero vector and is represented by For example, the velocity vector of a stationary object or the acceleration vector of an object moving with uniform velocity are null vectors.
It implies vectors of the same magnitude but opposite in direction.
Negative vector
Vectors having equal magnitude and same direction are called equal vectors.
If any two vectors are collinear, then one can be expressed in terms of the other. = (where λ is a constant)
Collinear vector
If two or more vectors start from the same point, then they are called co-initial vectors.
Here A, B, C, D are co-initial vectors.
Q2.
Sol:
Explanation:
Three (or more) vectors are called coplanar vectors if they lie in the same plane or are parallel to the same plane. Two (free) vectors are always coplanar.
Coplanar VectorsNote: If the frame of reference is translated or rotated the vector does not change (though its components may change).
Orthogonal unit vectors are unit vectors that are perpendicular to each other. When two or three unit vectors are at right angles, they are referred to as orthogonal unit vectors.
Orthogonal unit vectors
Localised Vectors are vectors that have a fixed starting point.
For example, consider a vector , this vector can be considered as a free vector because it doesn't matter where we draw this vector.
However, the vector is localised. It has a specific origin (the starting point) and it points towards a certain direction.
Localised vector
Non-localised Vectors are vectors that do not have a fixed starting point.
For example, the velocity vector of a particle moving along a straight path is considered a non-localised vector. Here, initial point is not fixed.
Non-localised vector
A displacement vector shows how much and in which direction an object has moved from its initial position to its final position within a certain time.
It is represented by the straight line connecting the starting and ending points, and it does not depend on the actual path taken by the object.
Q3.
Q4.
The resultant vector of two or more vectors is defined as the single vector that has the same effect as all the vectors combined.
There are two main cases:
The scalar product of two vectors is equal to the product of their magnitudes and the cosine of the smaller angle between them.
It is denoted by .(dot). The scalar or dot product of two vectors is a scalar.
A . B = AB cos θ
Properties of Scalar Product:
Q5. Find the cross product of two vectors a and b if their magnitudes are 5 and 10 respectively. Given that angle between then is 30°.
Sol:
The vector product of two vectors is equal to the product of their magnitudes and the sine of the smaller angle between them. It is denoted by X (cross).
Vector cross product
Properties of Vector Product:
The following results can be established:
Q6.
Sol:
Here is the summarised table of different types of vectors:
102 videos|411 docs|121 tests
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1. What is the difference between scalar and vector quantities? |
2. How do you calculate the dot product of two vectors? |
3. What is the significance of the cross product of two vectors? |
4. Can you give an example of a scalar quantity and a vector quantity? |
5. What are some common applications of vectors in physics? |
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