Products of Vectors

The product of vectors is the set of all possible vectors in a space that do not overlap. The product is the set of all possible vectors that are one-to-one, two-to-one or any other set that is not the union of two single vectors. The vectorial product is the force that brings a point in space in contact with another point in space. For example, the shape of the triangle is the product of the three vectors (the x, y and z axes) and the triangle is a product of the three vectors.

Types of Products of Vectors

There are two types of products of vectors: scalar and vector products.

Scalar Product of two Vectors

The scalar product of two vectors is the dot product of their components. The scalar product of two vectors can be used to transform a vector into a scalar that represents its value. In other words, the scalar product of two vectors is a scalar value representing the sum of each vector’s elements.

Scalar Product formula = A•B

                        = |A| |B| Cosθ

Properties of Scalar Products

1. a·b is a scalar quantity.

2. If the dot product to two quantities is zero, it means that they are perpendicular to each other. This is because Cosθ = 0 means θ=90°.

3. If the dot product of two quantities equals the product of their magnitudes, it means the two vectors are parallel. This is because Cosθ = 1 means θ=0°.

4. Similarly, if the dot product of two vectors equals the negative of the product of their magnitudes, it implies that they are antiparallel. This is because, for Cosθ= -1, the value of θ will be 180°.

5. While solving scalar products, 

i·i = j·j = k·k = 1 and i·j = i·k = j·k = 0 because of the same reasons as mentioned above.

6. Cosθ = a·b

                       |a||b|

7. Dot products (scalar products) are commutative in nature, that is, a·b = b·a.

8. They also follow the distributive property, that is, a·(b+c) = a·b + a·c.

Applications of Scalar Products

1. The dot product of force and displacement helps determine the work done by a system.

2. The scalar product helps determine if two vectors are orthogonal or not. If their dot product comes out to be zero, this would imply that they have an angle of 90° between them and are orthogonal to each other.

3. Scalar products help find the angle between two vectors and help determine their positions.

Vector Product of Two Vectors

The Vector product is a type of scalar field that can be used to describe the motion of matter. It is a field that includes both space-time and matter. The vector product of two vectors is a new vector field that describes the motion of a body. It is the second-order tensor of the space-time metric, with a contribution from the metric itself.

Vector Product formula = A×B

                        =|A| |B| Sinθ

Properties of Vector Product

1. The cross-product of two vectors is always a vector quantity.

2. If the cross product of two vectors is zero, it implies that the angle between them is 0 and they are parallel to each other.

3. While solving vector products, i×i = j×j = k×k = 0 and i×j = k, j×k=i, k×i = j. Also, i × k = -j, j×i = -k , k×j = -i. This is because cross product does not follow the commutative law of multiplication.

4. Sinθ = | a × b |

                        |a| |b|

5. Cross product does not follow the commutative law of multiplication, and hence, 

a×b = -(b×a).

6. However, a vector product follows the distributive law of multiplication; hence, a×(b+c) = a×b + a×c.

Applications of Cross Product of Two Vectors

1. Given a point and a direction, the product of the two vectors is the slope of the line through the point and the direction.

2. The product of vectors is a matrix that shows how a vector is projected on a plane if any.

3. Products of vectors help to find the volume of the parallelepiped, which has a formula a·(b×c).

4. It is used in finding the area of a parallelogram a × b and the area of triangle ½ (a×b).

5. It helps find the triple cross product and the angle between two vectors.

6. The torque acting on a body is calculated by taking the force’s cross product and the rotation radius.

Conclusion 

There are basically two types of products of vectors: scalar and vector. The scalar product of two vectors is the dot product of their directions. The scalar product of two vectors with unit magnitude is the dot product of the two vectors. The vector product is illustrated using the Cartesian product of two vectors. The resulting vectors are of the same length but can be obtained by different methods. The vector product is a special case of the cross product, which allows for the vector product to be obtained through the product of two vectors.