[Physics Class Notes] on Difference Between Scalar and Vector Pdf for Exam

The two important classifications of a quantity are scalar and vector. The main difference between scalar and vector quantity is that the scalar quantity is the one that is simply related to the magnitude of any quantity. The vector quantity is determined by both the magnitude as well as the direction of the physical quantity. 

 

Considering the dependency on the physical quantities and the direction, they can be classified into two categories – vector and scalar.

 

The concept of Physics is based on Mathematics. Several concepts and notions that we study in physics mostly rely on Mathematics. Scalar and vector quantities are the two mathematical quantities that explain the motion of a body.

 

For a common person, the two terms are the same, but there is a huge difference between them in physics. So, have a glance at the information below for your better understanding.

Definition of Scalar Quantity

The term scalar quantity is defined as a quantity containing only one element in the number field associated with the unit of measurements, such as a degree or meter. It is a quantity that refers only to size or magnitude, that is defined by a numerical value with a measuring unit. 

 

Algebra rules can be applied to link the scalar quantities, like scalars, multiplied, or subtracted similarly as numbers. However, for the scalar quantity, the operation is possible only for the quantities having a similar measurement unit.

Definition of Vector Quantity

Vector quantity is defined as the mathematical quantity that considers magnitude and direction as two independent characteristics to describe it. Here, magnitude represents the size of the quantity with absolute value. In contrast, direction represents the side i.e., north, east, south, west, north-east, etc.

 

The triangle law of addition is followed by vector quantity. The vector quantity represented with the help of an arrow placed over or next to the symbol denotes a vector.

Key Differences Between Scalar and Vector Quantities

The following points are important, so far as the scalar and vector difference is concerned:

1. The quantity that has magnitude is known as the scalar quantity. Whereas the vector quantity considers both magnitude and direction to describe its physical quantity.

2. One dimensional quantity can be explained by scalar quantities, for example, a speed of 35 km/h is a scalar quantity. Whereas multidimensional quantities can be described with the help of vector quantities, for example  increase and decrease in temperature.

3. When there is only a change in the magnitude, the scalar quantity changes, as per the vector quantity, changes in both magnitude and direction are required.

4. Scalar quantities follow the ordinary algebra rules like addition, multiplication, and subtraction, while vector quantity follows vector algebra rules for performance of operations.

5. Also, a scalar quantity can divide another scalar quantity, however, two vector quantities cannot be divided.
   

Scalar Product of Two Vectors

There are two ways in which vectors can be multiplied – scalar product and vector product. 

 

The scalar of two vectors can be achieved by moving the part of one vector in the other direction and multiplying it times the magnitude of the other vector. The scalar product is also known as inner product or dot product of vectors A and B are shown below:

A·B=|A||B|cos(θ)

 

Where,

| A | is magnitude of vector A,

 

| B | is magnitude of vector B, and

 

θ is an angle made by the two vectors

 

Here, θ is maximum if the direction of motion of an object is along the direction of plane or surface. However, it is zero when an object travels in a direction perpendicular to that of the surface. The outcome of a scalar product of two vectors is a scalar quantity.

 

For vectors given by their parts: A= (A_x,A_y,A_z) and B= (B_x,B_y,B_z), the scalar product is given by:

 

A·B = A_xB_x + A_yB_y + A_zB_z

 

Note that if θ = 90°, then cos(θ) = 0, and therefore we can state that two vectors, with magnitudes not equal to 0, are perpendicular if and only if their scalar product is equal to 0.

 

The cosine and therefore, the angle between the two vectors can be found with the help of scalar product.

 

cos(θ)= [frac {A.B}{|A| |B|}]

Properties of Scalar Product

1. A·B = B·A

2. A· (B + C) = A·B+ A·C

3. The value of the angle between two vectors is equal to the scalar product of the vectors divided by the product of their lengths.

4. If the vectors are perpendicular to each other, then the scalar product is equal to zero.

5. The scalar product of two vectors is maximum when the vectors are parallel and in the same direction.

6. If the vectors are antiparallel or in the opposite direction, then the scalar product is minimum.

7. The self dot product is defined as = A.A. In this product, the angle between the vectors is equal to zero degrees.

8. The scalar product of orthogonal unit vectors is given as:

i.j = j.k = k.i = 0 as these vectors are perpendicular to each other.

9. If the angle between the orthogonal unit vectors is zero degrees, then the scalar product is described as

i.i = j.j = k.k =1.

10.  If the angle between the vectors is acute, the scalar product is positive. This product is negative if the angle is obtuse. 
    

Vector Product of Two Vectors

The vector product is also known as the cross product. The resultant vector is perpendicular to the other two vectors. The outcome of the vector product is a vector quantity that can be determined using the Right-hand rule. The cross product of the two vectors A and B is given as

 

A x B = |A| |B| sin θ nˆ

 

Where,

|A| = magnitude of vector A

 

|B| = magnitude of vector B

 

θ = angle between the vectors A and B

 

nˆ = unit vector perpendicular to the plane containing the two vectors

 

The angle between the two vectors can be calculated as

θ = sin^-1 (A x B/ |A| |B|)

 

In the cartesian system, the vector product is written as

A = (ai + bj + ck)

 

B = (xi + yj + zk)

 

A x B = (ai + bj + ck) x  (xi + yj + zk)

 

A x B = ax(i × i) + ay(i × j) + az(i × k) + bx(j × i) + by(j × j) + bz(j × k) + cx(k × i) + cy(k × j) + cz(k × k)

 

A x B = ax(0) + ay(k) + az(-j) + bx(-k) + by(0) + bz(i) + cx(j) + cy(-i) + cz(0)

 

A x B = (bz – cy)i + (cx – az)j + (ay – bx)k

Properties of Vector Product

1. The vector product does not follow the commutative law.

A x B ≠ B x A

But, A x B = (-B) x A

2. The cross product shows the distributive property.

A x (B + C) = A x B + A x C

3. If the vectors are perpendicular to each other, the vector product will be maximum.

4. The cross product will be equal to zero if the two vectors are parallel or antiparallel to each other.

5. The self product of any vector is equal to zero.

6. The self product of orthogonal unit vectors is described as:

i x i = j x j = k x k = 0

7. The two perpendicular unit vectors will generate another unit vector.

i x j = k, j x k = i, k x i = j

Solved Example

1. Calculate the magnitude of the displacement if Ajay walks 10m north, 3m west, 5m south and 12m east.

Ans. Displacement is a vector quantity, therefore, the direction in which Ajay travelled will be positive or negative. 

Distance travelled along Y-axis (north is positive and south is negative)

Y = 10m – 5m = 5m

Distance travelled along X-axis (east is positive and west is negative)

X = -3m + 12m = 9m

Displacement will be calculated using the Pythagoras theorem

D² = X² + Y²

D² = 9² + 5²

D² = 81 + 25

D² = 106

D = 10.30m