Multiplication of Vectors

Vectors are objects in mathematics that have both magnitude and direction associated with them. The size of the vector can be determined based on its magnitude. It is depicted as a line with an arrow attached to it; the arrow indicates the direction in which the vector will move. If the magnitude and direction of two vectors are identical to one another, then the two vectors are considered to be identical. This indicates that if we translate a vector to a new position without rotating it, the vector that we end up with is the same vector that we started with. If we translate a vector without rotating it, the result is the same vector.

In order to complete a task that involves three dimensions, a binary vector operation is carried out. A type of vector multiplication known as a cross-product involves multiplying two vectors that are distinct from one another in their nature or type. It is possible to multiply two or more vectors by using either the cross product or the dot product. When you multiply two vectors together, you get another vector as the result. The resultant vector is also referred to as the vector product, which is another name for the cross product of two vectors. The final vector is located on the same plane as the two vectors that were used to create it.

Dot Product

A mathematical expression that represents the projection of one vector onto another is the dot product. Consider the situation where we have and the dot product of and is simply the projection of onto the vector of interest.

Examine this diagram to see what happens when we find the dot product of Vector A and B. and. We multiply the magnitude of the vector component B by the vector component A’s vector component along the direction of B.

A.B = (Acos θ).B = ABcos θ

As a result, the dot product of vectors A and B (A.B) is simply the product of the two vectors’ magnitudes multiplied by the cosine of the angle between them.

Some properties of Dot product

The dot product is often referred to as the scalar number because the final product is a scalar number. The following are some important properties of the scalar product to remember when working with it:

•The scalar product is commutative in the following way:

A.B = B.A

•The scalar product has the distributive property:

A.(B+C) = B.(A+C)

•When you take the scalar product of two perpendicular vectors, the result is always 0 (because cos90° is 0).

Cross product 

The cross product formula is used to calculate the area between any two vectors. The cross product formula determines the magnitude of the resultant vector, which is the area of the parallelogram spanned by the two vectors.

A^→ × B^→ = |a^→| |b^→| sin θn^ 

A^→ × B^→ = ia₂b₃ – a₃b₂ + ja₁b₃ – a₃b₁ + k(a₁b₂ – a₂b₁) 

Cross product of perpendicular vectors

The magnitude product, which is equal to the cross product of two vectors, is used to calculate the area of a rectangle with sides X and Y. When two vectors are perpendicular to one another, the cross product formula is:

θ = 90°

As we know sin 90° = 1 then,

X^→ × Y^→ = X^→.Y^→sin sinθ

X^→× Y^→= X^→.Y^→sin sin90°

Which is equal to the rectangle’s area

As a result, the perpendicular vectors’ cross product becomes

X^→× Y^→= X^→.Y^→

Conclusion

The term “cross product” refers to the operation of multiplying two vectors together. It is a binary vector operation that takes place in three dimensions. The cross product of the two vectors results in a third vector that is perpendicular to the two vectors that were originally presented. The area of the parallelogram that is formed between the two points is used to calculate its magnitude, while the right-hand thumb rule is used to calculate its direction. Because the results of the cross product of vectors is a vector quantity, it is also referred to as a vector product, which is another name for the cross product of two vectors. Take, for instance, the process of turning a bolt with a spanner: The length of the spanner is one unit of the vector. Another vector is the direction in which we apply force to the spanner in order to either tighten or loosen the bolt. The twisting motion that is produced is in a direction that is orthogonal to both vectors.