How to Calculate the Dot Product: Step-by-Step Guide
Geometric Definition of Dot Product
Let’s know the geometric definition of a dot product:
The scalar product of two vectors is known as the dot product.
The dot product is a scalar number obtained by performing a specific operation on the vector components.
The dot product is only for pairs of vectors having the same number of dimensions.
The symbol that is used for representing the dot product is a heavy dot.
This dot product is extensively in Physics as well as in Mathematics.
Here we are going to know about dot product distributive, the geometric meaning of dot product, geometric definition of dot product, properties of scalar and vector product, dot product commutative proof, distributive law of dot product, properties of dot product of two vectors, scalar product associative, vector product is not commutative, dot product algebraic definition, dot product geometrical definition, dot product of vector properties.
Concept of Dot Product
The concept of dot product states that any two vectors can be multiplied for getting the scalar quantity. It is used for getting the product. It is giving the products of two vectors or more vectors in two dimensions or more dimensions.
The geometric definition of the dot product says that the dot product between two vectors a and b is given as: a⋅b = |a||b|cos θ, where θ is the angle between two vectors a and b. In Mathematics, this formula is generally used for understanding the properties of the dot product. A formula for the dot product in terms of the vector components will make it easier to calculate the dot product between any two given vectors.
Dot Product - Geometrical Definition
The Dot Product of Vectors is written as a.b=|a||b|cosθ.
Where |a|, |b| are said to be the magnitudes of vector a and b and θ is the angle between vector a and b.
If any two given vectors are said to be Orthogonal, i.e., the angle between them is 90 then a.b = 0 as cos 90 is 0.
If the two vectors are parallel to each other the a.b =|a||b| as cos 0 is 1.
Dot Product - Algebraic Definition
The Dot Product of Vectors is written as
Dot product of vector - An example
Let there be two vectors [6,2,-1] and [5,-8,2]
a.b = (6)(5) + (2)(-8) + (-1)(2)
a.b = 30 - 16 - 2
a.b = 12
Let there be two vectors |a| equals 4 and |b| equals 2 and θ equals 60
a.b equals |a||b|cos 60
a.b equals 4.2 cos60
a.b equals 4.
Properties of Dot Product
(au + bv) · w equals (au) · w + (bv) · w, where a and b are known to be scalars
Below is the list of properties of the dot product:
u · v equals |u||v| cos θ
u · v equals v · u
u · v equals 0 when u and v are orthogonal.
0 · 0 equals 0
|v|2 equals v · v
a (u·v) equals (a u) · v
(au + bv) · w equals (au) · w + (bv) · w
The Formula for Dot Product
As the first step, we may see that the dot product between standard unit vectors, that is, the vectors i, j, and k of length one, and they are parallel to the coordinate axes.
The standard unit vectors in 3 dimensions. The standard unit vectors in 3 dimensions, i, j, and k are length one vectors that point parallel to the x-axis, y-axis, and z-axis respectively. Since the standard unit vectors are said to be orthogonal, we can immediately conclude that the dot product between a pair of distinct standard unit vectors is always zero: i⋅j equals i⋅k equals j⋅k equals 0.
The dot product between a unit vector and itself can be easily computed. In this case, the angle is zero, and cos θ = 1 as θ = 0. Given that the vectors are all of length one, the dot products are i⋅i = j⋅j = k⋅k equals to 1.
Since we know the dot product of unit vectors, we can simplify the dot product formula to, a⋅b = a1b1 + a2b2 + a3b3.
Solved Examples
Question 1) Calculate the dot product of a = (-4,-9) and b = (-1,2).
Solution: Using the following formula for the dot product of two-dimensional vectors, a⋅b = a1b1 + a2b2 + a3b3.
We calculate the dot product to be
= -4(-1) - 9(2)
= 4 - 18
= -14.
Question 2) Calculate the dot product of a = (-2,-4) and b = (-1,2).
Solution: Using the following formula for the dot product of two-dimensional vectors, a⋅b = a1b1 + a2b2 + a3b3
We calculate the dot product to be
= -2(-1) - 4(2)
= 2 - 8
= -6.
FAQs on Dot Product: Meaning, Formula, and Examples
1. What is the dot product of two vectors, as per the CBSE syllabus?
The dot product, also known as the scalar product, is a way to multiply two vectors. As per the CBSE Class 12 Maths syllabus, it can be defined in two equivalent ways:
- Geometrically: It is the product of the magnitudes of the two vectors and the cosine of the angle (θ) between them. The formula is a · b = |a| |b| cos(θ).
- Algebraically: It is the sum of the products of their corresponding components. For two vectors a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, the dot product is a · b = a₁b₁ + a₂b₂ + a₃b₃.
2. How is the dot product calculated for vectors in their component form (i, j, k)?
To calculate the dot product for two vectors in component form, you multiply their corresponding i, j, and k components and then add the results. For vectors A = Aₓi + Aᵧj + A₂k and B = Bₓi + Bᵧj + B₂k, the formula is:
A · B = (Aₓ * Bₓ) + (Aᵧ * Bᵧ) + (A₂ * B₂)
This works because the dot product of identical unit vectors (like i · i) is 1, while the dot product of different unit vectors (like i · j) is 0.
3. What are the most important properties of the scalar (dot) product?
The dot product has several key properties that are important for solving problems in vector algebra:
- Commutative Property: The order of vectors does not matter. a · b = b · a.
- Distributive Property: The dot product is distributive over vector addition. a · (b + c) = a · b + a · c.
- Scalar Multiplication: A scalar multiple can be grouped with either vector: (λa) · b = a · (λb) = λ(a · b).
- Self-Multiplication: The dot product of a vector with itself gives the square of its magnitude: a · a = |a|².
4. Why is the dot product called a 'scalar' product and what does its value represent?
The dot product is called a scalar product because the result of the operation is always a scalar (a single number with magnitude but no direction), not another vector. The value of the dot product a · b has a significant geometric meaning: it represents the product of the magnitude of vector a and the magnitude of the projection (or shadow) of vector b onto vector a.
5. What is the fundamental difference between the dot product and the cross product?
The primary difference lies in their output and geometric interpretation. The dot product of two vectors results in a scalar value and is used to find the angle between vectors or the projection of one onto another. In contrast, the cross product of two vectors results in a new vector that is perpendicular to both original vectors and is used to find the area of a parallelogram or a vector normal to a plane.
6. How is the dot product used to find the projection of one vector onto another?
The dot product is essential for calculating vector projections, a key topic in the CBSE syllabus. The projection of vector a onto a non-zero vector b is a vector that represents the 'shadow' of a along the direction of b. Its formula is derived using the dot product:
Projection of a onto b = ((a · b) / |b|²) * b
The scalar component of this projection, which is its length, is simply (a · b) / |b|.
7. What does it mean if the dot product of two non-zero vectors is exactly zero?
If the dot product of two non-zero vectors a and b is zero (i.e., a · b = 0), it means the vectors are orthogonal, or perpendicular to each other. This is because for a · b = |a||b|cos(θ) to be zero, the term cos(θ) must be zero. This only happens when the angle θ between the vectors is 90° or 270°.
8. What is a practical, real-world example where the dot product is applied?
A classic real-world application of the dot product is in physics for calculating Work Done. Work, a scalar quantity, is the dot product of the vector quantities Force (F) and Displacement (d). The formula is W = F · d = |F| |d| cos(θ). This correctly calculates that the work done is maximised when force is applied in the direction of displacement and is zero if the force is perpendicular to the displacement.
