We got dot product in scalar form whereas cross product in vector form...

Dot Product:
The dot product, also known as the scalar product, is an operation between two vectors that produces a scalar value. It is denoted by a dot (·) or by multiplying the vectors without any operator in between. The dot product measures the similarity or the projection of one vector onto another.

Properties of the Dot Product:
1. Commutative Property: The dot product of vectors A and B is equal regardless of the order in which they are multiplied.
2. Distributive Property: The dot product distributes over vector addition.
3. Scalar Multiplication Property: The dot product is linear with respect to scalar multiplication.

Calculation:
The dot product of two vectors A and B is calculated by multiplying their corresponding components and summing the results. If A = [a1, a2, a3] and B = [b1, b2, b3], then the dot product (A · B) is given by:
A · B = (a1 * b1) + (a2 * b2) + (a3 * b3)

Applications:
1. Finding the angle between two vectors.
2. Calculating work done in physics.
3. Determining the projection of one vector onto another.
4. Evaluating vector lengths and magnitudes.

Cross Product:
The cross product, also known as the vector product, is an operation between two vectors that produces a vector perpendicular to both. It is denoted by a cross (×) or by using the "cross" operator. The cross product measures the area of the parallelogram formed by the two vectors.

Properties of the Cross Product:
1. Anticommutative Property: The cross product of vectors A and B changes sign when the order of multiplication is reversed.
2. Distributive Property: The cross product distributes over vector addition.
3. Not Associative: The cross product is not associative, which means (A × B) × C is not equal to A × (B × C).

Calculation:
The cross product of two vectors A and B is calculated by multiplying their corresponding components using a specific formula. If A = [a1, a2, a3] and B = [b1, b2, b3], then the cross product (A × B) is given by:
A × B = [ (a2 * b3) - (a3 * b2), (a3 * b1) - (a1 * b3), (a1 * b2) - (a2 * b1) ]

Applications:
1. Calculating torque in physics.
2. Determining the direction of a magnetic field due to a current-carrying wire.
3. Finding the normal vector to a plane.
4. Solving problems related to rotation and angular momentum.

Differences:
1. Result Type: The dot product yields a scalar, while the cross product yields a vector.
2. Dimension: The dot product is defined for vectors of any dimension, while the cross product is only defined for vectors in three-dimensional space.
3. Orthogonality: The dot product measures the similarity or projection of vectors, while the cross product measures the perpendicularity or area between vectors.
4. Commutativity: The dot product is commutative,