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The Scalar Product | Physics Class 11 - NEET PDF Download

Velocity, displacement, force and acceleration are different types of vectors. Is there a way to find the effect of one vector on another? Can vectors operate in different directions and still affect each other? The answer lies in the Scalar product. An operation that reduces two or more vectors into a Scalar quantity! Let us see more!

What is a Scalar Quantity?

Before we learn about the Scalar product of two vectors, let’s refresh what we have already learned about the difference between a vector quantity and a scalar quantity. Scalar quantity is one dimensional and is described by its magnitude alone.  For example, distance, speed, mass etc.
Vector quantities, on the other hand, have a magnitude as well as a direction.  For example displacement, velocity, acceleration, force etc.  Some vector quantities are effectively the directional values of their corresponding scalar quantities. For example,  displacement, effectively, is the distance in a particular direction.  Vector quantities are expressed with an arrow above the appointed symbols. For example, displacement is expressed as The Scalar Product | Physics Class 11 - NEET The following image explains the difference between distance, a scalar quantity, and displacement, a vector quantity:
The Scalar Product | Physics Class 11 - NEET

In the above diagram, if we take the square AA1A2B and assume it’s side as 5m, then the object travelling the red path has travelled a distance of 15 meters.  But, the displacement of the said object is only 5 meters in the direction of A to B. Similarly, if we assume that the same object has finished the traversing the red path in one second, then, the object’s speed is 15 m/s. But, the velocity of the said object is only 5 m/s.
Now, in Physics, from time to time, we need to multiply two vector quantities. Some of these multiplications require a scalar product. For example, Work is a scalar quantity and is a product of Force and Displacement. Here, we will learn how to derive a scalar quantity as a product of two vectors, and, how these multiplications hold various laws of mathematics.

Scalar Product of Two Vectors

Let’s consider two vector quantities A and B. We denote them as follows:
The Scalar Product | Physics Class 11 - NEET

Their scalar product is A dot B.  It is defined as:
A.B = |A| |B| cosθ
Where, θ is the smaller angle between the vector A and vector B. An important reason to define it this way is that |B|cosθ is the projection of the vector B on the vector A. The projections can be understood from the following images:
The Scalar Product | Physics Class 11 - NEETThe Scalar Product | Physics Class 11 - NEET

Since, A(Bcosθ) = B(Acosθ), we can say that
A.B = B.A
Hence, we say that the scalar product follows the commutative law. Similarly, the scalar product also follows the distributive law:
A.(B+C) = A.B + A.C
Now, let us assume three unit vectors, i, j and k, along with the three mutually perpendicular axes X, Y and Z respectively.
The Scalar Product | Physics Class 11 - NEET

As cos (0) = 1, we have:
The Scalar Product | Physics Class 11 - NEET
Also since the cosine of 90 degrees is zero, we have:
The Scalar Product | Physics Class 11 - NEET
These two findings will help us deduce the scalar product of two vectors in three dimensions. Now, let’s assume two vectors alongside the above three axes:
The Scalar Product | Physics Class 11 - NEET
The Scalar Product | Physics Class 11 - NEET
So their scalar product will be,
The Scalar Product | Physics Class 11 - NEET
Hence,
A.B = AxBx + AyBy + AzBz  
Similarly, A2 or A.A =
The Scalar Product | Physics Class 11 - NEET
In Physics many quantities like work are represented by the scalar product of two vectors. The scalar product or the dot product is a mathematical operation that combines two vectors and results in a scalar. The magnitude of the scalar depends upon the magnitudes of the combining vectors and the inclination between them.

Solved Examples For You

Q: Let’s find the angle between force and the displacement; where, F = (2i + 3j + 4k) and d = (4i + 2j + 3k).
Solution: We already know that,  A.B = AxBx + AyBy + AzBz
Hence, F.d = Fxdx + Fydy + Fzdz = 2*4 + 3*2 + 4*3 = 26 units 
Also, F.d = F dcosθ
Now, F² = 2² + 3² + 4² = √29 units 
Similarly, d² = 4² + 2² + 3² = √29 units 
Hence, F d cosθ = 26 
Then, cosθ = 26/(F d) = 26/(√29 × √29) = 26/29, which gives cosθ = 0.89 or θ = cos-1(0.89) 

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FAQs on The Scalar Product - Physics Class 11 - NEET

1. What is a scalar quantity?
A scalar quantity is a type of physical quantity that has only magnitude and no direction. It can be described by a single real number or value. Examples of scalar quantities include mass, temperature, time, and speed.
2. What is the scalar product?
The scalar product, also known as the dot product, is a mathematical operation between two vectors that results in a scalar quantity. It is calculated by multiplying the magnitudes of the two vectors and the cosine of the angle between them. The formula for the scalar product of two vectors A and B is A · B = |A| |B| cosθ.
3. What are some examples of scalar quantities?
Some examples of scalar quantities are: - Mass: The amount of matter in an object. - Temperature: The measure of the average kinetic energy of particles in a substance. - Speed: The rate at which an object is moving. - Time: The measurement of the duration between two events. - Distance: The length between two points.
4. How is the scalar product useful in physics?
The scalar product is useful in physics for various calculations and concepts. It helps determine the angle between two vectors, find projections of one vector onto another, calculate work done by a force, and determine whether two vectors are perpendicular or parallel to each other. It also plays a role in calculating the magnitude of a vector and solving problems related to forces and motion.
5. What is the difference between a scalar quantity and a vector quantity?
The main difference between a scalar quantity and a vector quantity is that scalar quantities only have magnitude, while vector quantities have both magnitude and direction. Scalar quantities can be represented by a single real number, whereas vector quantities require both magnitude and direction. Examples of vector quantities include displacement, velocity, acceleration, and force.
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