Dot Product Formula Proof Properties and Solved Examples
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 in Vectors Definition and Applications
1. What is the dot product in mathematics?
The dot product (also called the scalar product) is an operation that takes two vectors and returns a single number (scalar).
For vectors a and b, it is defined as:
a · b = |a||b| cosθ
Where:
- |a| and |b| are the magnitudes of the vectors
- θ is the angle between them
2. What is the formula for the dot product?
The formula for the dot product of two vectors in component form is a · b = a₁b₁ + a₂b₂ + a₃b₃ (in 3D).
For example, if:
- a = (a₁, a₂)
- b = (b₁, b₂)
a · b = a₁b₁ + a₂b₂
This formula is used for coordinate-based calculations in algebra and physics.
3. How do you calculate the dot product step by step?
To calculate the dot product, multiply corresponding components and then add the results.
Steps:
- Write both vectors in component form.
- Multiply corresponding components.
- Add the products.
If a = (2, 3) and b = (4, 5),
a · b = (2×4) + (3×5) = 8 + 15 = 23.
4. What does the dot product represent geometrically?
Geometrically, the dot product represents the product of the magnitudes of two vectors and the cosine of the angle between them.
a · b = |a||b| cosθ
This means:
- If θ = 0°, the dot product is maximum (vectors point in same direction).
- If θ = 90°, the dot product is 0 (vectors are perpendicular).
- If θ = 180°, the dot product is negative.
5. How do you find the angle between two vectors using the dot product?
You can find the angle between two vectors using the formula cosθ = (a · b) / (|a||b|).
Steps:
- Compute the dot product a · b.
- Find the magnitudes |a| and |b|.
- Substitute into the formula.
- Take cos⁻¹ to find θ.
6. When is the dot product equal to zero?
The dot product is zero when the two vectors are perpendicular (orthogonal).
From the formula:
a · b = |a||b| cosθ
If θ = 90°, then cos90° = 0, so:
a · b = 0
This property is often used to test whether two vectors are perpendicular in coordinate geometry.
7. What is the difference between dot product and cross product?
The main difference is that the dot product gives a scalar, while the cross product gives a vector.
Key differences:
- Dot product: a · b = |a||b| cosθ (result is a number).
- Cross product: a × b = |a||b| sinθ n̂ (result is a vector).
- Dot product measures alignment.
- Cross product measures area and direction perpendicular to both vectors.
8. Can you give an example of a dot product calculation?
Yes, here is a simple example of a dot product calculation.
Let a = (1, −2, 3) and b = (4, 0, −1).
Compute:
a · b = (1×4) + (−2×0) + (3×−1)
= 4 + 0 − 3 = 1.
The final dot product is 1, which is a scalar value.
9. What are the properties of the dot product?
The dot product satisfies several important algebraic properties.
Key properties:
- Commutative: a · b = b · a
- Distributive: a · (b + c) = a · b + a · c
- Scalar multiplication: (ka) · b = k(a · b)
- Self product: a · a = |a|²
10. What is the dot product of a vector with itself?
The dot product of a vector with itself equals the square of its magnitude, that is a · a = |a|².
Example:
If a = (3, 4), then:
- a · a = (3×3) + (4×4)
- = 9 + 16 = 25
