Table of Contents
What is a dot product?
The dot product is an introductory system of combining two vectors. It’s used to know the direction of two vectors. The dot product of two vectors is also known as the attendant and it’s a scalar volume.
The dot product is signified by a.b and also read as “A dot B ”
The formula of dot product:
The formula of the dot product is
a . b = | a | . | b | Cos θ
- a & b are two vectors.
- |a| & |b| are the magnitudes of vectors a & b.
- “θ” is the angle between a & b.
The dot product, also known as the scalar product or inner product, is a mathematical operation that is used in linear algebra and vector calculus. It is defined as the product of the magnitude of two vectors and the cosine of the angle between them. The result of the dot product is a scalar value, which represents the projection of one vector onto the other.
A dot product calculator is a tool that calculates the dot product of two vectors. This can be useful in a variety of applications, such as determining the angle between two vectors, finding the projection of one vector onto another, and computing the length of a vector.
To use a dot product calculator, simply input the components of the two vectors you want to find the dot product of. The calculator will then return the dot product, which can be used in various calculations and analyses.
It’s important to note that the dot product is only defined for two vectors in the same dimensional space, and that the dot product of two orthogonal (perpendicular) vectors is equal to zero. This property can be used to determine whether two vectors are orthogonal, as well as to find the angle between them.
In summary, the dot product is a useful mathematical operation that can be used in a variety of applications, and a dot product calculator is a handy tool for quickly finding the dot product of two vectors. Whether you’re a student studying linear algebra or a professional working in engineering, physics, or computer graphics, a dot product calculator can be a valuable tool in your mathematical toolkit.
How to find dot product?
The dot product is defined as the product of the magnitude of the two vectors and the cosine of the angle between the two given vectors. If a and b are two vectors then the dot product is given as follows:
a . b = | a | . | b | Cos θ
- a & b are two vectors.
- |a| & |b| are the magnitudes of vectors a & b.
- “θ” is the angle between a & b.
Example:
Find the dot product of two vectors having magnitudes of 6 units and 7 units, and the angle between the vectors is 60°.
Solution:
The magnitudes of the two vectors are |a| = 6, |b| = 7, and the angle between the vectors is θ = 60°.
The dot product of the two vectors is: a. b = | a |.| b| cos θ°
= (6) (7) cos 60°
= (6) (7) (1/2)
= (3) (7)
= 21
Algebraic Expression:
A concise expression has been formulated for easy calculation. And that is:
Where,
a is first vector
b is second vector
Σ is summation; it is the total of all the dimensional dot product values
n is the dimension number (n=1,2,3,……..)
Here, a1, a2, a3………an are all the values only concerning vector a, but in different dimensions, like
a1 is for x-axis,
a2 is for y-axis
a3 is for z-axis
And so on.
The same is the case for vector b. Values of both a and b vectors are used to get the final product.
Applications of Dot Product in Real Life:
The dot product has numerous applications in various fields, including physics, engineering, computer science, and more. In this article, we will explore some of the exclusive applications of the
dot product.
• Testing the Orthogonality
• Image and Signal Processing
• Solving Systems of Linear Equations
• Calculating Work Done by a Force
• Testing for Collinearity
• Finding Projections
Testing the Orthogonality
The very first one in my list of top 6 applications of the dot product is that you can test the orthogonality using dot products. According to the definition of dot products, two vectors are orthogonal,
only, and, only if, their dot product is zero. You can use this property in many applications such as finding the normal vector to a plane, testing for perpendicularity in geometry, etc.
Image and Signal Processing
Yup, it’s true. You can extensively use dot products for image and signal processing.
The dot product is used in image compression algorithms to reduce the amount of data and size to represent the image.
Not to mention, it is also used in audio processing to determine the similarity between two audio signals. For example, in this article, all the images that I am using are compressed by using dot
products. Well, of course, I personally have not used it to compress the size of an image, the tools I use did.
Solving Systems of Linear Equations
You can use the dot product to find the unknown vector in the system of linear equations. A system of linear equations can be represented in a matrix form i.e
ax = b
where,
a = matrix of coefficients
x = vector of variables
b = vector of constants
Here, you can use the dot product to find the dot product of each row of (A) with the vector (x). Equating these dot products with the corresponding elements of (b) will give you a set of equations that can be easily solved to find the values of variables.
Calculating Work Done by a Force
One of the most important application of dot product in physics is in calculating work done by a force. You can use the dot product of force and displacement to calculate the work done by a force. The formula for work done by a force is:
w = F . d
where,
w = Work done
F = Force vector
d = displacement vector
Testing for Col-linearity
As per the definition, two vectors are col-linear, only, and, only if, one vector is a scalar multiple of the other. So, how can you test if this theory is correct or not?
You can use the dot product of two vectors to check whether two vectors are col-linear or not. If the dot product of two vectors is equal to the product of their magnitudes, then the vectors are col-linear.
Finding Projections
Last but not least one in my list of top 6 dot products applications is that with the help of dot products, you can easily find the projection of one vector onto another vector.
Mathematically, the projection of a vector (U) on a vector (V) is given by:
proj v(u) = ((uv) / (|v|^2))v
Some Other Uses of Dot Product in Real Life
Apart from the above-mentioned ones, here are few others.
• Determining the magnitude of a vector.
• Calculating the eigenvalues and eigenvectors of a matrix in linear algebra.
• Finding the intersection of two lines in geometry.
• Determining the direction of maximum change in a function.
• Determining the velocity and acceleration of an object in kinematics, etc.
What is dot product (scalar product)?
The dot product, also called scalar product, is a measure of how closely two vectors align, in terms of the directions they point. The measure is a scalar number (single value) that can be used to compare the two vectors and to understand the impact of repositioning one or both of them. The dot product is obtained by performing mathematical calculations on
the vector properties.
A vector is a quantity that has both direction and magnitude (the vector’s length). Figure 1 shows two vectors (a and b) on a twodimensional Cartesian plane. Vector a has a magnitude of 8 and is at a 115-degree angle from the x-axis (moving counter clockwise). Vector b has a magnitude of 10 and is at a 45- degree angle from the x-axis. The angle between the two vectors — represented by the Greek letter theta (θ) — is 70 degrees, which is calculated by subtracting 45 degrees from 115 degrees.
If the magnitude of two vectors and the angle between them is known, it is easy to calculate the dot product. The dot product is represented by using a dot between the two vector references, in this case, a and b, as shown in the following formula:
a • b
The full equation for finding the dot product is a bit more involved. This entails multiplying the magnitude of vector a by the magnitude of vector b and then multiplying the product by the cosine (cos) of the angle between the vectors, as shown in the following equation:
a • b = |a| × |b| × cos(θ)
The vertical bars indicate that these values are the vector’s magnitude. Sometimes double bars are used instead of single bars. If the values from Figure 1 are plugged into the equation, calculating the dot product for these two vectors can be quickly done, as shown in the following equation:
a • b = 8 × 10 × cos(70 degrees)
a • b = 8 × 10 × 0.342
a • b = 27.36
The angle’s cosine has been rounded down to three decimal places, so the final product (27.36) is only an approximation, although it’s a close one.
Three-Dimensional Dot Product
A similar approach can be used to calculate the dot product for vectors in a three dimensional space. For this, modify the formula as follows to incorporate the third dimension (represented by the z-axis):
a • b = (ax × bx) + (ay × by) + (az × bz)
How to calculate three dimensional dot product?
Problem: The coordinates for the x-axis, y-axis, and z-axis for the vectors a and b are given as – (4,8,10) and (9,2,7).
Solution: Substitute the coordinate values in the formula for the two-dimensional dot product.
Formula – a · b = ax × bx + ay × by + az × bz
Substituting, we get
a . b = (4 × 9) + (8 × 2) + (10 × 7)
On solving, we get
a . b = (36) + (16) + (70)
a . b = 36 + 16 + 70
a . b = 122
Therefore, the dot product of vectors a and b is 122.
What are the properties of Dot Product?
Here are the key properties of the dot product:
Commutativity
The dot product is commutative, which means that the order of the vectors does not matter:
a · b = b · a
Distribution over vector addition:
The dot product distributes over vector addition, allowing you to perform the dot product separately on each vector and then add the results:
(a + b) · c = a · c + b · c
Linearity:
The dot product exhibits linearity with respect to scalar multiplication. This means that you can pull out scalar multiples from the dot product:
(k * a) · b = k * (a · b) = a · (k * b), where k is a scalar.
Orthogonality:
Two vectors are orthogonal (perpendicular) if and only if their dot product is zero. This property is useful in determining if vectors are perpendicular or not:
a · b = 0 if a and b are orthogonal.
Non-negativity:
The dot product of a vector with itself is always non-negative, and it is zero if and only if the vector is the zero vector:
a · a ≥ 0, with equality if and only if a is the zero vector.
Conclusion:
Vector is a quantity that has magnitude as well as direction. Few mathematical operations can be applied to vectors such as addition and multiplication. The multiplication of vectors can be done in dot product.The dot product of two vectors is the sum of the products of their corresponding components. It is the product of their magnitudes multiplied by the cosine of the angle between them. A vector’s dot product with itself is the square of its magnitude.