Vector | Physics, Definition | Note For Science

Definition of Scalar quantity:

The physical quantities which have only magnitude and don't have direction are called scalar quantity.

Example: Mass, Volume, Density, Time, Speed etc.

 

Definition of Vector Quantity:

The physical quantities which have both magnitude and as well as direction and obey the laws of vector addition are called vector quantity.

Example: Momentum, Velocity, Force etc

Differences between scalar quantity and vector quantity.

Scalar quantity:

1. Scalar quantity have only magnitude.

2. Scalar quantities change if their magnitude changes.

3. Scalar quantities can be added according to the ordinary laws of algebra.

Vector quantity:

1. Vector quantity have both magnitude and direction.

2. Vector quantities change if either their magnitude, direction or both changes.

3. Vector quantities can be added only by using special laws of vector algebra.

Representation of a Vector

A vector quantity can be represented by a straight line with an arrow head over it. The length of the line gives the magnitude and the arrow head gives the direction.

Definition of Position Vector

Position vector is the vector which gives position of an object with reference to the origin of a co-ordinate system.

Representation of Position Vector

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Let us consider the motion of an object is at $XY$ plane with origin $O$ at an object is at point $P$ at any instant $t$ as shown in figure. Then $\vec{OP}$ is the position vector of the object at point $P$.

The position vector provide us two information about the object:

1. Position vector tells the straight line distance of the object from the origin.

2. Position vector tells the direction of the object with respect to the origin.

Definition of Displacement Vector

Displacement vector is the vector which tells how much and in which direction an object has changed its position in a given time interval.

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Let us consider an object moving in the $XY$ plane. Suppose it at point $P$ at any instant $t$ and at point $Q$ at any later instant $t'$ as shown in figure. The vector $\vec{PQ}$ is the displacement vector of the object in time $t$ to $t'$.

Modulus of a Vector

The modulus of a vector means the length or magnitude of that vector. It is scalar quantity.

Modulus of a vector $\vec{A} = | \vec{A} | = A$

Definition of Unit Vector

It is a vector of unit magnitude drawn in the direction of a given vector.

A unit vector in the direction of a given vector is found by dividing the given vector by its modulus.

Thus a unit vector in the direction of $\vec{A}$ is given by $$ \hat{A} = \frac{\vec{A}}{A} $$

The unit vector along positive directions of X-axis, Y-axis and Z-axis are denoted by $\hat{i}$, $\hat{j}$ and $\hat{k}$ respectively.

Resolution or Components of a Vector

The components of a vector $\hat{A}$ are its projections on the respective co-ordinate axes and are called rectangular components.

$A_x$, $A_y$, $A_z$ denote the magnitude of $\vec{A}$ along X, Y and Z axis.

Hence, $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$.

Definition of Null Vector

A vector whose initial point and final point coincides called a null vector and as denoted by $\vec{0}$.

Equality of a Vector

Two non-zero vector are said to be equal if they are in the same direction and have one and the same absolute value.

Explanation of Dot product and Cross product of two vectors.

Dot Product of two vectors

The scalar or dot product of two vectors $\vec{A}$ and $\vec{B}$ is defined as the product of two magnitudes of $\vec{A}$ and $\vec{B}$ and cosine of the angle $\theta$ between them.

Thus $$\vec{A}.\vec{B} = | \vec{A} | | \vec{B} |cos\theta = ABcos\theta$$

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Cross product of two vectors

The vector or cross product of two vectors is a vector whose magnitude is the product of magnitudes of individual vectors and sine smaller between them. Direction of the resultant vector will be perpendicular to the plane containing two the vectors. If we consider the angle between $\vec{A}$ and $\vec{B}$ is $\theta$

then,

$$ \vec{A}×\vec{B} = ABsin\theta\hat{n}$$

Where, $\hat{n}$ is a unit vector perpendicular to the plane $\vec{A}$ and $\vec{B}$ and its direction is given by right hand thumb rule.

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