## 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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CHECK OUT: Cross Product Calculator

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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

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

- Select the type of vector (2 dimensional or 3 Dimensional)
- Select the type of input from dropdown (i, j, k format or angle and magnitude format)
- IMPORTANT -> Do not leave rows blank between two vector details. Multiple vectors needs to be added continuously without skipping any row.
- 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).