A dot product calculator is a convenient tool for anyone who needs to solve multiplication problems involving vectors. Rather than manually computing the scalar product, you can simply input the required values (two or more vectors here) on this vector dot product calculator and it does the math for you to find out the dot (inner) product.
How to use the dot product calculator?
This is dot product calculator is a very simple tool that’s easy to understand. As long as you have the required values, you can use it to make automatic calculations. Here are the steps to follow for this matrix dot product calculator:
- First, input the values for Vector a which are X1, Y1, and Z1.
- Then input the values for Vector b which are X2, Y2, and Z2.
- After inputting all of these values, the dot product solver automatically generates the values for the Dot Product and the Angle Between Vectors for you.
What is the dot product of two vectors?
Despite the convenience of the dot product calculator which is also known as a dot product of two vectors calculator or a matrix dot product calculator, you may want to perform the calculation by hand. To do this, you must draw both of the vectors and separate them by an angle.
Then if you try figuring out the image of the scalar product, you’ll find out that you need to multiply two parts namely the projection of one of the vectors towards the direction of the second vector along with that vector. Since these parts are parallel, the result you get is the product of the lengths of both parts.
Although there are two ways to perform this operation, you still get the same result. In other words, the dot product comes from the multiplication of the length of vectors projected in the direction of one of these vectors.
How do you calculate the dot product?
As aforementioned, there are two kinds of vector multiplication namely the scalar or dot product represented as “•” and the cross product represented as “×.” The biggest difference between these products is that the product of a dot operation is always a single number and that of a cross operation is always a vector.
To calculate the dot product without a vector dot product calculator, let’s assume that we’ll perform our calculations in a 3D space. In such a case, you can write each of the vectors using 3 components:
a = [a₁, a₂, a₃] b = [b₁, b₂, b₃]
Geometrically speaking, the dot product is the product of the magnitudes of vectors multiplied by the value of the cosine of the angle between the vectors. You can express this with the following equation:
a•b = |a| * |b| * cosα
If you aren’t sure about the magnitude of a vector or how to perform the calculation, you’d be better off using a dot product of two vectors calculator. But if you want to put in the effort to calculate manually, let’s continue.
If you have a 90˚ angle between your two vectors, then you will always get a scalar product equal to zero no matter what the magnitudes of the vectors are. Similarly, if you have a 0˚ angle which means that you have collinear vectors, you can find the dot product by only multiplying the multitudes.
Algebraically speaking, the dot product refers to the sum of the products of the components of vectors. Therefore, if you have a vector with 3 components, your dot product formula would be:
a•b = a₁ * b₁ + a₂ * b₂ + a₃ * b₃
In any space which have more than 3 dimensions, add more terms to your summation. But if you’re multiplying vectors in a 2D space, remove the 3rd term in your dot product formula.
You can also use the dot product calculator to solve for the angle between two given vectors wherein the cosine is the ratio of the vectors’ magnitudes and the scalar products. For this, you need this formula:
cosα = a•b / (|a| * |b|).
If you’re wondering how this dot product solver works, you need to follow some steps. To help you understand this better, let’s use an example:
- Choose the values for vector a. Let’s use a = [4, 5, -3].
- Choose the values for vector b. Let’s use b = [1, -2, -2].
- Solve for the product of each vector’s first components. For this example, the solution looks like this: 4 * 1 = 4.
- Solve for the product of each vector’s middle components. For this example, the solution looks like this: 5 * (-2) = -10.
- Solve for the product of each vector’s third components. For this example, the solution looks like this: (-3) * (-2) = 6.
• Now add all of the value together to solve for the dot product: 4 + (-10) + 6 = 0.
What is dot product used for?
A dot product has several applications including:
- Proving the law of cosines using the dot product. When you draw a triangle using 3 vectors, you can write the formula as c = b – a. If you need to solve for c2, you can expand this equation as c² = (b-a)•(b-a) = b•b – b•a – a•b + a•a = a² + b² – |b| * |a| * cosa – |a| * |b| * cosa = a² + b² – 2 * |a| * |b| * cosα. This, incidentally, is how you can prove the cosine law.
- Using it to find out whether two given vectors are perpendicular to each other.
- Defining different kinds of physical quantities as dot products.
- Working as a dot product of displacement and force.
- Using power as a dot product of velocity and force.
- Using magnetic or electric flux as a dot product of a magnetic or electric field along with the surface which it flows through.
- Using magnetic potential energy as a dot product of a magnetic field and magnetic moment.