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!
“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.
Our vector multiplication calculator uses the following equation to find the results of the vector scalar multiplication:
$$ a.b = |a| |b| cosθ $$
Where;
With respect to the above equation, it is confirmed that the dot product for perpendicular vectors is always zero because cos (90) = 0.
The dot product formulas in two and three dimensions are different due to unequal number of components in vectors.
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:
$$ a.b = \lt a_{1}.b_{1} + a_{2}.b_{2} \gt $$
Now, if we are having vectors in three dimensional space, then their components will be three and hence the dot product is defined as:
$$ A = \lt a_{1}, a_{2}, a_{3} \gt $$
$$ B = \lt b_{1}, b_{2}, b_{3} \gt $$
$$ a.b = \lt a_{1}.b_{1} + a_{2}.b_{2} + a_{3}.b_{3} \gt $$
$$ θ = cos^{-1}\dfrac{a.b}{|a| |b|} $$
$$ |a| = \sqrt{a_{1}^{2} + a_{2}^{2}} $$
$$ |a| = \sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2}} $$
How to do dot product for a couple of vectors given as under:
$$ A = <3, 5> $$
$$ B = <4, 8> $$
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.
Our calculator is very simple to use! It functions to provide you with dot product calculations if given the following inputs:
What To Do?
If you chose “Vector Components”:
If you chose “Magnitudes & Angle”:
What You Get!
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.
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.
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
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