Dot Product Calculator

Dot Product Calculator

This dot product calculator is used to find the dot product for the given vector components. It can also be used to determine the magnitude and angle between the vectors and shows step-by-step dot product and related calculations for better understanding. 

What Is The Dot Product?

The dot product multiplies the corresponding components of each vector and adds the products together. This multiplication (product) results in a scalar value.

Geometrically, it is the product of the Euclidean magnitudes and the cosine of the angle between the vectors. 

The dot product tells you how much of the first arrow "line up" with the second arrow:

• If both the arrows point in the same direction, then it means that the dot product is a positive number
• If the arrows are pointing in the opposite direction, then the result of the dot product is a negative number
• When the arrows are perpendicular (or orthogonal), the dot product is zero

Dot Product Formula:

\(\vec u.\vec v = (u_1v_1\ +\ u _2v_2\ +\ u_3v_3...u_nv_n)\)

This is the most common method when the two vectors are in cartesian coordinates. The dot product of two vectors is based on the projection of one vector onto another and depends on the magnitude of both vectors. It is beneficial when you need to measure the angle between two vectors and want to find the position of a vector relative to the coordinate axes. 

If you have the magnitude and the angle between the vectors then you can find dot product as:

\(\vec u.\vec v = |u||v|cos(\theta)\)

Where:

• \(\vec{u}\cdot\vec{v}\) represents the dot product of vectors
• \(\|\vec{u}| \ and\ |\vec{v}|\) are the magnitudes of the vectors \(\vec{u}\ and\ \vec{v}\)
• \(\theta\) is the angle between \(\vec{u}\ and\ \vec{v}\)

Angle Between Vectors:

\(\cos(\theta) = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| \, |\vec{v}|}\)

• When the dot product of two vectors gives a positive value then the angle between the vectors is considered acute
• But if it is negative then the angle is obtuse
• If the result of the dot product is zero then the vectors are perpendicular to each other 

How To Calculate The Dot Product of Two Vectors?

There are three methods to find the dot product of the vectors, which are:

Method #1: Using Magnitudes and Angle Formula

Example#1:Suppose you have two vectors \(\vec u\ and\ \vec v\) with magnitudes \(|\vec u| =\ 5, |\vec v| =\ 7\), and the angle between the vectors is 45 degrees. Calculate the dot product of vectors.

Solution: 

Given that:

• Magnitude of \(\ |\vec u| =\ 5\)
• Magnitude of \(\ |\vec v| =\ 7\)
• The angle (\(\theta\)) = 45°

Put values in the dot product formula:

\(\vec u\cdot \vec v = (5) (7) cos(45°) ≈ 24.74 \) (rounded to two decimal places).

Method #2: Using Vector Coordinates

Example #2: Calculate the dot product of vectors u = (3, 4, 1) and v = (1, 4, -2) and also find the angle between the vectors.

Solution:

Identify components:

\(\ u_1 = 3, u_2 = 4, u_3 = 1\)

\(\ v_1 = 1, v_2 = 4, v_3 = -2\)

By adding values in the formula: 

\(\ (3\ \times\ 1) + (4\ \times\ 4) + (1\ \times\ -2) = 3 +16 + (-2)\)

Sum the products:

 3 +16 + (-2) = 17

The dot product of vectors u and v is 17

Now to find the angle, we need to get the magnitude of both vectors, So

The magnitude of vector \(\vec u\):

\(\ |\vec{u}| = \sqrt{(3)^{2} + (4)^{2} + (1)^{2}}\)

\(\ |\vec{u}| = \sqrt{9+ 16 + 1}\)

\(\ |\vec{u}| = \sqrt{26}\)

\(\ |\vec{u}| = \ 5.09\)

The magnitude of vector \(\vec v\):

\(\ |\vec{v}| = \sqrt{(1)^{2} + (4)^{2} + (-2)^{2}}\)

\(\ |\vec{v}| = \sqrt{2+ 16 + 4}\)

\(\ |\vec{v}| = \sqrt{22}\)

\(\ |\vec{v}| = \ 4.69\) (rounded to two decimal places)

Find the angle:

\(\theta =\ cos^{-1} \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| \, |\vec{v}|}\)

\(\theta =\ cos^{-1} \frac{17}{\ 5.09\ \times\ 4.69}\)

\(\theta =\ cos^{-1} (0.7121)\) (rounded to three decimals)

\(\theta ≈\ 44.59\ deg\)

Method #3: Using Vector Dot Product Calculator:

An online calculator can also be used for calculating the scalar product of vector components. It is one of the fastest ways to perform the multiplication of vectors.

These are the steps that you only need to follow to calculate the vector’s dot product:

• Step #1: Enter vector coordinates/components, or magnitudes & angles (if available)
• Step #2: Click on “Calculate” to get the result from the dot product vector calculator

FAQ’s:

Why Do We Calculate Dot Product?

The dot product of two vectors helps to understand specific relationships between vectors that represent physical quantities like displacement, force, and velocity. It considers both their direction and magnitude. 

• The positive result means the vectors are aligned in the same direction
• A negative result indicates that the vectors are in opposite directions
• The zero value shows that the vectors are perpendicular to each other

When it comes to calculations, using the dot product calculator provides instant assistance to simplify problems that involve the multiplication of vectors. 

What's The Difference Between A Dot Product and A Cross Product?

• The result of a dot product is a scalar quantity and it only represents the magnitude
• The cross-product results in a vector quantity and it indicates both magnitude and direction

References:

From the authorized source of Wikipedia: Dot Product.