Dot product vs. cross product of two vectors
Dot and cross products of two vectors are fundamental mathematical operations useful for understanding and dealing with vector behaviors. They find their application in various fields, including physics, mathematics, and computer graphics.
In this Answer, we will be making a comparison between these two operations to understand the major differences between them.
Dot product
The dot product of two vectors is a mathematical operation in which two vectors result in a
: The resultant scalar.
Note: To study about the dot product in detail, we can refer to this Answer.
Cross product
The cross product of two vectors is a mathematical operation that results in a vector
: The resulting vector that is always perpendicular to both and .
Note: To study about the cross product in detail, we can refer to this Answer.
Comparison
Let's compare the dot and cross product of two vectors now.
Dot product | Cross product |
It is also known as inner product or scalar product. | It is also known as vector product. |
The resultant is a scalar. | The resultant is a vector. |
The resultant doesn't have a direction. | The resultant has a direction. |
The geometric formula includes the cos of the angle: v1 . v2=∣v1∣∣v2∣cosθ here v1 and v2 are vectors. | The geometric formula includes the sin of the angle: v1 × v2=∣v1∣∣v2∣sinθ here v1 and v2 are vectors. |
The dot product of two parallel vectors (angle equals 0) is the maximum. | The cross product of two parallel vectors (angle equals 0) is the minimum. |
The dot product of two perpendicular vectors (angle equals 90) is the minimum. | The cross product of two perpendicular vectors (angle equals 90) is the maximum. |
It helps determine if two vectors are perpendicular (we check if their dot product is 0). | It helps determine if two vectors are parallel (we check if their cross product is 0). |
It is commutative: v1 . v2 = v2 . v1 here v1 and v2 are vectors. | It is anti-commutative: v1 × v2 = - v2 × v1 here v1 and v2 are vectors. |
The dot product of a vector with itself is equal to square of its magnitude: v · v = |v|^2. | The cross product of a vector with itself is equal to a zero vector: v × v = 0. |
Conclusion
These differences help us in deciding which operation we should use. Mainly, the dot product helps us calculate work, energy, and projections between two vectors. On the other hand, the cross product helps determine perpendicularity between two vectors and calculate angular momentum and magnetic fields in physics.
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