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Introduction to Unit Vector Dot Product

A unit vector dot product is the result of the scalar multiplication of two unit vectors.

A unit vector is a vector having magnitude equal to 1. In a three dimensional coordinate plane, the unit vectors along x, y and z axes are hati, hatj and hatk respectively. The following diagram represents unit vectors in a three dimensional coordinate plane:


  Unit Vector


When two vectors are multiplied by the method of scalar multiplication, their product is called a dot product. Dot products are the products of scalar multiplication. For two vectors`vecA`  and `vecB` , their dot product is defined as

`vecA*vecB = A*B*costheta` , where

`A =` magnitude of `vecA`   `= |vecA|`

`B =`  magnitude of `vecB = |vecB|`

`theta`  = angle between `vecA`  and `vecB`



Calculation of Unit Vector Dot Product


All unit vectors have magnitude 1 and the unit vectors in a three dimensional coordinate plane, that is, the unit vectors `hati` , `hatj`  and `hatk` , are perpendicular to each othe. Thus, the angle between two unit vectors is `theta`  = 90 degrees. Thus, the resultant of the dot products of unit vectors can be predetermined.

As described above, the scalar (dot) product of two vectors is the product of their individual magnitudes and the cosine of the angle between the two vectors. Thus, the dot product of two unit vectors can be obtained as,

`hati*hatj = i*j*costheta` , where

`i = 1`

`j = 1` , and

`theta = 90`

`hati * hatj = 1*1*cos90`

`cos 90 = 0`

Thus,

`hati*hatj = 0` .

Similarly, `hatj * hatk = 0 and hati * hatk = 0` .

Thus, the dot product of two different unit vectors is 0.

Now we will calculate the dot product of a unit vector with itself.

`hati * hati == i * i * cos theta`

we know that i = 1 and `theta`  = 0 degrees, since the angle between same vectors is 0.

therefore `hati * hati = 1 * 1 * cos0`

`= 1 (Since cos0 = 1)`

Thus, the self product of unit vectors by scalar multiplication is 1.

Thus, `(hati)^2 = (hatj)^2 = (hatk)^2 = 1` .


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Difference between Unit Vector Dot Product and Cross Product


The dot product of a unit vector is different from the cross product of a unit vector. This is because the dot product is a scalar product and the cross product is a vector product. Thus, the resultant of a dot product is a scalar quantity with no direction, whereas the resultant of a cross product is a vector quantity.


The vector product of two vectors `vecA`  and `vecB`  is given by

`vec(Vab) = A*B*sin(theta)*hatn` , where

`A = magnitude of vecA = |vecA|` ,

`B = magnitude of vecB = |vecB|` ,

`theta = "angle" between vecA and vecB` ,

`hatn = a`  unit vector perpendicular to the plane in which `vecA` and `vecB` are present.


In the above definition, hatn is a vector, hence the cross product of unit vectors is a vector quantity, whereas their dot product is not a vector quantity.

Sin `theta`  has different values than cos theta (except when `theta`  = 45 degrees) and thus the magnitude of cross product of unit vectors is also different from that of the dot product of unit vectors.


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