Dot Product Calculator

Dot Product Calculator

This dot product calculator is used to find the dot product for the given vector components. it can additionally be used to decide the significance and attitude between the vectors and shows step-via-step dot product and associated calculations for better information.

what's the Dot Product?

The dot product multiplies the corresponding additives of each vector and provides the goods collectively. This multiplication (product) results in a scalar value.

Geometrically, it's miles the made of the Euclidean magnitudes and the cosine of the angle among the vectors.

The dot product tells you the way a good deal of the primary arrow "line up" with the second one arrow:

• If both the arrows factor within the identical path, then it method that the dot product is a high quality quantity
• If the arrows are pointing inside the contrary route, then the end result of the dot product is a negative range
• while the arrows are perpendicular (or orthogonal), the dot product is 0

Dot Product formula:

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

This is the most not unusual method when the two vectors are in cartesian coordinates. The dot made of two vectors is primarily based at the projection of one vector onto another and relies upon on the importance of both vectors. it's miles useful when you need to measure the perspective between vectors and need to discover the position of a vector relative to the coordinate axes.

if you have the significance and the perspective between the vectors then you may 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:

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

• while the dot product of vectors gives a high quality value then the angle between the vectors is taken into consideration acute
• but if it's miles terrible then the attitude is obtuse
• If the end result of the dot product is zero then the vectors are perpendicular to every different

how to Calculate The Dot product of two Vectors?

There are 3 techniques to discover the dot made from the vectors, which can be:

Approach #1: using Magnitudes and angle components

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