Difference between dot product and cross product

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cross product is vector product as its direction is perpendicular to other variables whereas dot product is scalar one as its direction can be of any one variable

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Dot product is multiple of two vector quantity and resultant scalar quantity.It is equal to product between magnitude of two vectors and cos of the angle between them.dot product is multiple of two vector quantity and resultant is vector quantity.It is equal to product between magnitude of two vectors and sin of angle between them and a unit vector along resultant.A.B=A*B*cos​O where O is angle between A and BA*B=A*B*sinO*n where O is angle between A and B and n is unit vecor along resultant

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Difference between Dot Product and Cross Product

The dot product and cross product are two fundamental operations in vector algebra used to manipulate vectors in three-dimensional space. While both operations involve vectors, they have distinct properties and serve different purposes. Let's delve into the details of each operation and highlight their differences.

Dot Product:
The dot product, also known as the scalar product, is a binary operation that takes two vectors as input and produces a scalar as the result. It is denoted by the symbol '⋅' or by using the formula:

𝐴 ⋅ 𝐵 = |𝐴| |𝐵| cos(θ)

where 𝐴 and 𝐵 are vectors, |𝐴| and |𝐵| represent the magnitudes of the vectors, and θ is the angle between the two vectors.

Key points about the dot product:
- The dot product is commutative, meaning that 𝐴 ⋅ 𝐵 = 𝐵 ⋅ 𝐴.
- The result of the dot product is a scalar quantity, which means it has only magnitude and no direction.
- The dot product can be used to find the angle between two vectors since it involves the cosine of the angle.
- The dot product is used in various applications, such as calculating work done, determining projections, and finding the angle of incidence in physics.

Cross Product:
The cross product, also known as the vector product, is a binary operation that takes two vectors as input and produces a vector as the result. It is denoted by the symbol '×' or by using the formula:

𝐴 × 𝐵 = |𝐴| |𝐵| sin(θ) 𝑛

where 𝐴 and 𝐵 are vectors, |𝐴| and |𝐵| represent the magnitudes of the vectors, θ is the angle between the two vectors, and 𝑛 is a unit vector perpendicular to the plane containing 𝐴 and 𝐵.

Key points about the cross product:
- The cross product is not commutative, meaning that 𝐴 × 𝐵 = -𝐵 × 𝐴.
- The result of the cross product is a vector quantity, which means it has both magnitude and direction.
- The magnitude of the cross product is given by |𝐴 × 𝐵| = |𝐴| |𝐵| sin(θ), where θ is the angle between the two vectors.
- The direction of the cross product is determined by the right-hand rule, which relates the direction of the cross product to the direction of rotation from 𝐴 to 𝐵.
- The cross product is used in various applications, such as calculating torque, determining the normal vector to a plane, and finding the area of a parallelogram.

Summary:
In summary, the dot product is a scalar operation that yields a scalar result, while the cross product is a vector operation that produces a vector result. The dot product measures the similarity or alignment between two vectors, while the cross product measures the perpendicularity or rotational aspects between two vectors. Understanding the distinctions and applications of these operations is crucial in vector algebra and various fields of science and engineering.

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tq ..

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Diopter is the SI unit of focal length what will be the the CGS unit of it

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