Dot product - animation

Consider two vectors a and b, the angle between them being α. For the sake of simplicity we have represented them in two dimensions in the figure below:

The dot product or scalar product returns a scalar (that is, a number) and is given by:

From the expression above we can see that the dot product of two perpendicular vectors is 0.

The dot product can be used to find the projection of a vector onto another one. As an example we are going to find the projection of a onto b.

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The projection of a vector on a line can be found by drawing a line from the end of the vector perpendicular to the line onto which we want to project it. The resulting distance over the line is the projection ab:

Analytically it can be found by using the cosine function:

The direction defined by a vector is given by the unit vector ub:

And performing the dot product on a and ub:

Which is precisely the projection of a onto b.

The dot product is commutative:

From the previous results, it can be deduced:

When both vectors a and b are expressed in unit vector notation, as shown in the first figure, the dot product is given by:

Finally, we can find the angle between two vectors by using the dot product:

The dot product is used in Physics to define the work of a force.

In the animation below b’ represents b rotated 900. Since the cosine is the sine complement, the area of the parallelogram that vectors a and b’ span is the absolute value of the dot product a · b. You can move both vectors a and b to see their dot product.

Dot product (




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