What Is Dot Product (Scalar Product)?
The dot product also referred to as scalar product is a number (scalar quantity) obtained by performing a specific operation on the vector components. The dot product has meaning only for pairs of vectors having the same number of dimensions. Dot product number features prominently in many problems in physics and variants of it appear in an enormous number of mathematical areas.
In the two-dimensional Cartesian plane, vectors are expressed in terms of the x-coordinates and y-coordinates of their end points, assuming they begin at the origin (x, y)= (0,0). The dot product of two vectors is determined by multiplying their x-coordinates, then multiplying their y-coordinates and finally adding the two products.
What You Need To Know About Dot Product
- If the product of two vectors is a scalar quantity, the product is referred to as a scalar product or dot product.
- The dot product is obtained by multiplying the corresponding entries and then summing the products.
- If two vectors are perpendicular to each other, then their scalar product is zero.
- The scalar product obeys commutative law as A.B=B.A
- Dot product outcome does not specify direction.
- The dot product is used for the purpose of projection of one vector on another.
- In algebraic operations, the dot product takes two equal length sequences of numbers and gives a single number.
- Some of the applications of dot product include calculating distance of a point to a plane, calculating distance of a point to a line and calculating projection of a point.
- If the vectors are named “a” and “b” then the dot product is represented by “a. b.” This is equal to the magnitudes multiplied by the cosine of the angles. (A.B=AB Cos Ɵ).
Cross Product (Vector Product)
Cross product also referred to as Vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol. Given two linearly independent vectors and the cross product, is a vector that is perpendicular to both and thus normal to the plane containing them. If two vectors have the same direction (or have the exact opposite direction from one another i.e are not linearly independent) or if either one has zero length, then their cross product is zero.
The cross product has applications in various fields e.g it is used in computational geometry, physics and engineering. For example, cross product appears in the calculation of the distance of two skew lines (lines not in the same plane) from each other in three-dimensional space. In computational geometry of the planes, the cross product is used to determine the sign of the acute angle defined by three points.
What You Need To Know About Cross Product
- If the product of two vectors is a vector quantity then the product is referred to as vector product or cross product.
- Cross product can be described as a binary operation on two vectors in a three-dimensional space.
- If two vectors are parallel to each other, their vector product is zero.
- The vector or cross product does not obey commutative law, AXB≠BXA
- Cross product outcome specifies direction.
- Cross product is not used for the projection of one vector on another.
- The cross product results in a vector that is perpendicular to both the vectors that are multiplied and normal to the plane.
- Some of the applications of cross product are calculating distance of a point to a plane and calculating the specular light.
- In vectors “a” and “b”, the cross product is represented by “a X b’’. This is equal to the magnitudes multiplied by the sine of the angles and thereafter multiplied by “n”, a unit vector. (AXB=AB Sin Ɵ n)
Also Read: Difference Between Scalar And Vector Quantity
Difference Between Dot Product And Cross Product In Tabular Form
BASIS OF COMPARISON | DOT PRODUCT | CROSS PRODUCT |
Description | If the product of two vectors is a scalar quantity, the product is referred to as a scalar product or dot product. | If the product of two vectors is a vector quantity then the product is referred to as vector product or cross product. |
How It Is Obtained | The dot product is obtained by multiplying the corresponding entries and then summing the products. | Cross product can be described as a binary operation on two vectors in a three-dimensional space. |
Zero vector/scalar product | If two vectors are perpendicular to each other, then their scalar product is zero. | If two vectors are parallel to each other, their vector product is zero. |
Commutative Law | The scalar product obeys commutative law as A.B=B.A | The vector or cross product does not obey commutative law, AXB≠BXA |
Direction | Dot product outcome does not specify direction. | Cross product outcome specifies direction. |
Use | The dot product is used for the purpose of projection of one vector on another. | Cross product is not used for the projection of one vector on another. |
Result | In algebraic operations, the dot product takes two equal length sequences of numbers and gives a single number. | The cross product results in a vector that is perpendicular to both the vectors that are multiplied and normal to the plane. |
Application | Some of the applications of dot product include calculating distance of a point to a plane, calculating distance of a point to a line and calculating projection of a point. | Some of the applications of cross product are calculating distance of a point to a plane and calculating the specular light. |
Representation | If the vectors are named “a” and “b” then the dot product is represented by “a. b.” This is equal to the magnitudes multiplied by the cosine of the angles. (A.B=AB Cos Ɵ). | In vectors “a” and “b”, the cross product is represented by “a X b’’. This is equal to the magnitudes multiplied by the sine of the angles and thereafter multiplied by “n”, a unit vector. (AXB=AB Sin Ɵ n) |