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Dot product Calculator

What is a dot product of two vectors

A dot product is scalar multiplication of 2 vectors resulting in a scalar value when number of vectors multiplied are even.


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When number of vectors multiplied are even the resultant quantity will be a scalar value. Where as when the number of vectors multiplied are odd, you will get the result to be a scaled up vector quantity.

Physical significance of a dot product

If i, j, k are considered as unit vectors along X, Y and Z axes then dot product signifies that the X component of vector 1 has 0 contribution in Y and Z direction of vector 2.

Similarly the Y component of vector 1 has 0 significance in X and Z direction of vector 2. Likewise the Z component has 0 significance in the X & Y direction of vector 2.

Dot product formula

Consider the following example of how to find the dot product:

If two vectors are represented in terms of unit vectors i,j,k then the dot product formula is given as:-

V_{1}=a_{1} \hat{\imath}+b_{1} \hat{\jmath}+c_{1} \hat{k}

V_{2}=a_{2} \hat{\imath}+b_{2} \hat{\jmath}+c_{2} \hat{k}

V_{1} * V_{2}=a_{1} * a_{2}+b_{1} * b_{2}+c_{1} * c_{2}

If both the vectors are expressed in terms of magnitude and angle then the formula is given as:

V_{1}=\left\{A, \theta_{1}\right\}

V_{2}=\left\{B, \theta_{2}\right\}

Where A and B are the magnitudes of vector 1 and 2 respectively, theta 1 and theta 2 are the angles of vector 1 and vector 2 respectively.

V_{1} * V_{2}=A * B * \cos \left(\theta_{2}-\theta_{1}\right)

Unit vectors in dot products

Following rules are followed while taking dot products of unit vectors:

i * i = 1

i * j = 0

i * k = 0

j * j = 1

j * i = 0

j * k = 0

k * k = 1

k * i = 0

k * j = 0

From this we can see that the contribution individual components of both the vectors happen only in their respective axes directions.

How to use calculator

  1. Select the type of vector (2 dimensional or 3 Dimensional)
  2. Select the type of input from dropdown (i, j, k format or angle and magnitude format)
  3. IMPORTANT -> Do not leave rows blank between two vector details. Multiple vectors needs to be added continuously without skipping any row.
  4. When angle and magnitude format is selected for 2D case, you need to enter only one angle measured from X axis. For 3D case Angle 2 is dependent on angle 1 and it can only be greater than (90 – angle1).
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