Dot Product Calculator

Get the dot product of vectors with this advanced dot product calculator. With that, the tool also functions to provide you with the magnitudes of vectors and the angle between them. Moreover, get complete steps involved in the calculations with utmost accuracy.

Make your scalar product calculations be done in seconds!

What Is The Dot Product of Vectors?

“It is the product of two vectors in terms of their components”

The dot product of two vectors always yields a scalar and not a vector.

Dot Product Formula:

Our vector multiplication calculator uses the following equation to find the results of the vector scalar multiplication:

Basic: 

$$ a.b = |a| |b| cosθ $$

Where;

• |a| = Magnitude of Vector A
• |b| = Magnitude of Vector B
• θ = Angle between the Vectors

With respect to the above equation, it is confirmed that the dot product for perpendicular vectors is always zero because cos (90) = 0.

Component Wise:

The dot product formulas in two and three dimensions are different due to unequal number of components in vectors.

2D Plane:

If we have vectors A and B such that:

$$ A = \lt a_{1}, a_{2} \gt $$

$$ B = \lt b_{1}, b_{2} \gt $$

Their dot product is given as:

$$ a.b = \lt a_{1}.b_{1} + a_{2}.b_{2} \gt $$

3D Plane:

Now, if we are having vectors in three dimensional space, then their components will be three and hence the dot product is defined as:

$$ A = \lt a_{1}, a_{2}, a_{3} \gt $$

$$ B = \lt b_{1}, b_{2}, b_{3} \gt $$

$$ a.b = \lt a_{1}.b_{1} + a_{2}.b_{2} + a_{3}.b_{3} \gt $$

Angle Between The Vectors:

$$ θ = cos^{-1}\dfrac{a.b}{|a| |b|} $$

Magnitude of Vectors:

2D Plane:

$$ |a| = \sqrt{a_{1}^{2} + a_{2}^{2}} $$

3D Plane:

$$ |a| = \sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2}} $$

Dot Product Example:

How to do dot product for a couple of vectors given as under:

$$ A = <3, 5> $$ 

$$ B = <4, 8> $$

Solution:

Step 01: Magnitude of vector A

$$ |a| = \sqrt{a_{1}^{2} + a_{2}^{2}} $$

$$ |a| = \sqrt{3^{2} + 5^{2}} $$

$$ |a| = \sqrt{9 + 25} $$

$$ |a| =  5.83 $$

Step 02: Magnitude of vector B

$$ |b| = \sqrt{b_{1}^{2} + b_{2}^{2}} $$

$$ |b| = \sqrt{4^{2} + 8^{2}} $$

$$ |b| = \sqrt{16 + 64} $$

$$ |b| = \sqrt{80} $$

$$ |b| = 8.94 $$

Step 03: Dot Product

Here we have:

$$ a.b = \lt a_{1}.b_{1} + a_{2}.b_{2} \gt $$

$$ a.b = \lt 3.4 + 5.8 \gt $$

$$ a.b = \lt 12 + 40 \gt $$

$$ a.b = \lt 52 \gt $$

Step 04: Angle between Vectors

$$ θ = cos^{-1}\dfrac{a.b}{|a| |b|} $$

$$ θ = cos^{-1}\dfrac{52}{5.83*8.94} $$

$$ θ = cos^{-1}\dfrac{52}{52.1202} $$

$$ θ = cos^{-1}0.997 $$

$$ θ = 0.99^{o} $$

These calculations may be overwhelming. That is why using a vector dot product calculator will complete these steps in moments and save you a lot of time.

Working of Dot Product Calculator:

Our calculator is very simple to use! It functions to provide you with dot product calculations if given the following inputs:

What To Do?

• First, choose the method of calculations

If you chose “Vector Components”:

• Select dimensions i.e; 2D or 3D
• From the next list, select if you want to pursue calculations w.r.t coordinates or points
• Based on your selection, enter the values 
• Tap Calculate

If you chose “Magnitudes & Angle”:

• Enter the magnitudes of both the vectors
• Enter the angle between them
• Tap Calculate

What You Get!

• Dot product
• Magnnitudes of vectors 
• Angle between vectors 
• Complete steps

FAQs:

What Is The Difference Between Dot Product & Vector Product?

The dot product yields the scalar number. On the other hand, a vector product always gives you a vector that is orthogonal to the given ones.

What If The Dot Product Is Zero?

If the dot product of a couple of vectors is zero (0), it shows that both of them are exactly at a right angle to each other.

References:

Wikipedia: Dot product, Coordinate definition, Geometric definition, Scalar projection and first properties, Properties

Khan Academy: Vectors and notation, A better way to compute the dot product