Dot Product and Cross Product of Vectors in 3D Cartesian Coordinates6

In three-dimensional Cartesian coordinates, vectors are represented by ordered triplets of numbers, called components. The dot product and cross product are two important vector operations that are used in a variety of applications.

Dot Product

The dot product of two vectors a = (a1, a2, a3) and b = (b1, b2, b3) is a scalar quantity, denoted by a b, that is given by the following formula:```
a b = a1b1 + a2b2 + a3b3
```

The dot product has the following properties:* Commutative: a b = b a
* Distributive: a (b + c) = a b + a c
* Associative: (a b) c = a (b c)
* Scalar multiplication: r(a b) = (ra) b = a (rb), where r is a scalar

The dot product can be used to calculate the following:* Length of a vector: The length of a vector a is given by ||a|| = √(a a).
* Angle between two vectors: The angle θ between two vectors a and b is given by cos θ = (a b) / (||a|| ||b||).

Cross Product

The cross product of two vectors a = (a1, a2, a3) and b = (b1, b2, b3) is a vector quantity, denoted by a × b, that is given by the following formula:```
a × b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)
```

The cross product has the following properties:* Anti-commutative: a × b = -b × a
* Distributive: a × (b + c) = a × b + a × c
* Associative: (a × b) × c = a × (b × c)
* Scalar multiplication: r(a × b) = (ra) × b = a × (rb), where r is a scalar

The cross product can be used to calculate the following:* Area of a parallelogram: The area of a parallelogram spanned by two vectors a and b is given by ||a × b||.
* Volume of a parallelepiped: The volume of a parallelepiped spanned by three vectors a, b, and c is given by |a (b × c)|.

Applications

The dot product and cross product are used in a variety of applications, including the following:* Computer graphics: The dot product is used to calculate lighting and shading effects, and the cross product is used to calculate

2025-01-01

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