Vector & Scalar Quantities - Physics Class 11 - NEET PDF Download

Different physical quantities can be classified into the following two categories:

Scalars

Scalar quantity is a physical quantity that has magnitude only and no direction.

• It can be represented by a number along with an appropriate unit. 
  Example: Mass = 4 kg.
• Magnitude and unit are separate attributes: for mass = 4 kg, magnitude = 4, unit = kg.
• Scalar quantities follow the ordinary laws of algebra for addition, subtraction and multiplication by real numbers.
• Examples of scalar quantities: mass, speed, distance, time, area, volume, density, temperature, energy, work.

Vectors

Vector quantity is a physical quantity that has both magnitude and direction and follows the laws of vector addition.

Vectors

Example: Speed = 4 m/s (scalar); Velocity = 4 m/s towards north (vector).

• The magnitude (or absolute value) of a vector A is denoted by |A|.
• Examples of vector quantities: displacement, velocity, acceleration, force, momentum.

Difference between scalar and vector quantities is illustrated below:

Classification of Vectors

Polar Vectors

Vectors which have a definite starting point (point of application) are called polar or localized vectors. Examples: force, displacement, reaction, etc.

Polar Vectors

Axial Vectors

Vectors that arise from rotational effects and are directed along the axis of rotation are called axial vectors (also called pseudovectors).

Examples: Angular velocity, angular momentum, torque.

Axial Vectors

The direction of an axial vector is determined by the sense of rotation:

• For anti-clockwise rotation, the axial vector points upwards along the axis of rotation.
• For clockwise rotation, the axial vector points downwards along the axis of rotation.
• The direction is fixed by whether the rotation is anti-clockwise or clockwise (use right-hand rule conventions where applicable).

General Points Regarding Vectors

Representation of a Vector

Geometrically, a vector is represented by a directed straight line (an arrow). The length of the arrow is proportional to the magnitude and the arrowhead indicates the direction.

Mathematically, a vector may be written as A→, or in boldface as A, or using component notation.

For a vector in the XY-plane making an angle θ with the x-axis, the arrow as shown represents the vector.

Graphical example: To represent a force of 5 N acting 45° north of east, draw a directed line of appropriate scale making 45° with the east (x) axis.

The Angle Between Two Vectors (θ)

The angle between two vectors is the smaller angle formed when the vectors are brought to a common initial point by moving them parallel to themselves (tail-to-tail).

Unit Vector

A unit vector is a vector of magnitude unity that indicates only direction.

The unit vector in the direction of A is written as â or Â.

It is expressed as

Worked Example - Q1

Q1.
Sol:

Null (Zero) Vector

A null vector (or zero vector) has magnitude zero and no definite direction. It is usually denoted by 0→.

Examples: the velocity vector of a stationary object, acceleration of an object moving with constant velocity (if acceleration = 0).

Negative of a Vector

The negative of a vector has the same magnitude but opposite direction to the original vector. If A→ is a vector, -A→ points in the opposite direction.

Negative vector

Equality of Vectors

Two vectors are equal if they have equal magnitudes and the same direction. Equality does not depend on the position in space (for free vectors).

Collinear Vectors

Vectors are collinear if they lie along the same straight line or are parallel to the same line. If A→ and B→ are collinear, one can be written as a scalar multiple of the other: B→ = λ A→, where λ is a scalar.

Collinear vector

Co-initial Vectors

Vectors that have the same initial point are called co-initial vectors.

In the figure, A, B, C, D are co-initial vectors.

Worked Example - Q2

Q2.

Sol:

Coplanar Vectors

Three or more vectors are coplanar if they lie in the same plane or are parallel to the same plane. Any two free vectors are always coplanar.

Coplanar Vectors

Note: If the frame of reference is translated or rotated, the vector itself does not change, although its components relative to a coordinate system may change.

Orthogonal Unit Vectors

Orthogonal unit vectors are unit vectors that are mutually perpendicular. Typical Cartesian orthonormal basis vectors are î, ĵ, k̂ (or ^i, ^j, ^k).

Orthogonal unit vectors

Localised and Non-localised Vectors

Localised Vectors

Localised vectors have a fixed starting point (a specific point of application). Such vectors cannot be translated freely without changing the physical situation.

Example: A position vector from the origin to a particular point is localised.

The following vector has a specific origin and so is localised:

Localised vector

Non-localised Vectors

Non-localised (or free) vectors do not have a fixed starting point; they can be translated parallel to themselves without changing their physical effect.

Example: The velocity vector of a particle moving along a straight path can be represented as a non-localised vector when only magnitude and direction matter.

Non-localised vector

Position Vector

The position vector of a point P is the vector drawn from the origin O of a coordinate system to the point P. It locates the point in space relative to the chosen origin.

Displacement Vector

The displacement vector shows the change in position of an object. It is represented by the straight line joining the initial and final positions and is independent of the path taken.

Worked Questions - Q3 and Q4

Q3. 

Q4.

Multiplication of a Vector by a Scalar

When a vector A→ is multiplied by a scalar k, the resulting vector kA→ has magnitude |k||A| and direction:

• same as A→ if k > 0,
• opposite to A→ if k < />
• zero vector if k = 0.

Resultant Vector

The resultant of two or more vectors is a single vector which has the same effect as the combination of the given vectors.

There are two simple cases for two vectors A and B:

Case I: Two vectors in the same direction

• If vectors A and B act in the same direction then the resultant R has the same direction as A and B.
• The magnitude is R = A + B (sum of magnitudes).

Case II: Two vectors in opposite directions

• If vectors A and B act in opposite directions then the resultant R lies along the direction of the larger vector.
• The magnitude is R = |A - B| (the absolute difference of magnitudes).
• If B > A, the resultant is along B; if A > B, the resultant is along A.

Scalar (Dot) Product of Two Vectors

The scalar product of two vectors equals the product of their magnitudes and the cosine of the smaller angle between them.

It is denoted by a dot: A · B = AB cos θ. The result of a dot product is a scalar.

Properties of the Scalar Product

• Commutative: A · B = B · A.
• Distributive over addition: A · (B + C) = A · B + A · C.
• Scalar product of perpendicular vectors is zero: A · B = AB cos 90° = 0.
• Scalar product of parallel vectors equals the product of magnitudes: A · B = AB cos 0° = AB.
• Scalar product of a vector with itself is the square of its magnitude: A · A = A^2.
• In Cartesian coordinates: A · B = A_xB_x + A_yB_y + A_zB_z (where A_x etc. are components).

Q5. Find the cross product of two vectors a and b if their magnitudes are 5 and 10 respectively. Given that angle between them is 30°.

Sol:

Vector (Cross) Product of Two Vectors

The vector product of two vectors equals the product of their magnitudes and the sine of the smaller angle between them; it results in a vector.

Vector cross product

• The magnitude is |A × B| = AB sin θ.
• The direction is perpendicular to the plane containing A and B; its sense is given by the right-hand thumb rule.
• The vector product A × B is itself a vector (orthogonal to both A and B).

Properties of the Vector Product

• Not commutative: A × B = - (B × A).
• Distributive over addition: A × (B + C) = A × B + A × C.
• Cross product of parallel vectors is zero: A × B = 0 when vectors are parallel.
• Cross product of a vector with itself is zero: A × A = 0.
• Using a right-handed cyclic order of unit vectors, moving clockwise or anti-clockwise gives the corresponding third unit vector with sign conventions.

The following standard results are commonly used for unit vectors and component calculations:

Direction of Vector Cross Product

• For C = A × B, the vector C is perpendicular to the plane containing A and B.
• The direction is determined by the right-hand thumb rule: curl the fingers of your right hand from A to B through the smaller angle; the erect thumb then points in the direction of A × B.

Right Hand Thumb Rule

• Right Hand Screw Rule (Maxwell's screw rule): it is an alternative visual method to determine direction. Imagine turning a right-handed screw from A toward B; the direction in which the screw advances along the axis corresponds to the direction of the cross product or magnetic field.
• Turning direction (clockwise or counterclockwise) indicates the sense of rotation; the screw's axial motion gives the direction of the resultant vector (for current, magnetic field or angular momentum contexts).

Q6.

Sol: