Tanvesh
Tanvesh
Masters in Structural Engineering | Research Interest - Artificial Intelligence and Machine learning in Civil Engineering | Youtuber | Teacher | Currently working as Research Scholar at NIT Goa

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This cross product calculator can calculate cross producs of two or multiple vectors and also give the direction, magnitude and angle.

Cross Product Calculator

Cross product of two vectors

This calculator can be used to calculate the cross product of two or more vectors. The vectors can be specified in i,j,k format or angle and magnitude format.

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A cross product is a vector multiplication which will always result in another vector.

Cross product formula

Cross product of two vectors can be calculated by matrix method or direct formula, both of which will be discussed.

1) Cross product Matrix method

If vector V1 = a1 i + b1 j + c1 k 

and vector V2 = a2 i + b2 j + c2 k then cross product is given by the determinant of the below matrix.

\overline{V}_{1} \times \overline{V_{2}}=\left[\begin{array}{lll} i & j & k \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right]

alternatively this can be written as formula :

\overline{V_{1}} \times \overline{V_{2}}=\left(b_{1} c_{2}-b_{2} c_{1}\right) * \hat{\imath}-\left(a_{1} c_{2}-a_{2} c_{1}\right) * \hat{\jmath}+\left(a_{1} b_{2}-a_{2} b_{1}\right) * \hat{k}

2) Cross product of unit vectors method

Cross product calculator
Cross product calculator

The cross product of unit vectors can be calculated from the given circle. As it is seen i x j = k; j x i = -k; j x k = i; k x j = -i; k x i = j; i x k = -j.

Using these rules we can find the cross product of two vectors.

Right hand thumb rule for vectors

Right hand thumb rule for vectors (source: https://commons.wikimedia.org/wiki/File:Manoderecha.svg)

According to the right hand thumb rule if the finger curl in the direction of Vector 1 to Vector 2, then the direction of thumb gives the direction of cross product of two vectors.

Cross product magnitude

If two vectors seperated by angle theta represented as a parallelogram, then the cross product magnitude is given by the area of the parallelogram.

Cross product magnitude

The magnitude can also be given by the formula \text { Magnitude }=\left|V_{1}\right| *\left|V_{2}\right| * \sin \theta

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