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Three-dimensional vectors are fundamental in describing direction and magnitude in a 3D space. They are denoted by components along the x, y, and z axes, represented by unit vectors i, j, and k. Graphical representation in a coordinate system, matrix formulation, and operations such as dot product, magnitude calculation, and angle determination are crucial for their application in scientific and engineering fields.

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## Definition of 3D Vectors

### Mathematical objects in a three-dimensional space

3D vectors are mathematical objects used to describe direction and magnitude in a three-dimensional space

### Components and notation of 3D vectors

Unit vectors i, j, and k

Unit vectors i, j, and k correspond to the x, y, and z axes and are used to denote the components of a 3D vector

Component notation and example

3D vectors are typically denoted by their components along the unit vectors, such as A→=3i+2j+5k

### Graphical representation of 3D vectors

3D vectors can be graphically represented by plotting them in a three-dimensional coordinate system and labeling their terminal point

## Matrix Representation and Operations

### Matrix representation of 3D vectors

3D vectors can also be represented as column matrices with three entries, each corresponding to the vector's i, j, and k components

### Vector operations

Dot product

The dot product is a key operation involving 3D vectors, which quantifies the extent to which two vectors align in direction

Magnitude calculation

The magnitude of a 3D vector can be calculated using the three-dimensional Pythagorean theorem

Angle determination

The angle between two 3D vectors can be determined using the dot product and the magnitudes of the vectors

## Applications of 3D Vectors

### Importance in scientific and engineering contexts

3D vectors play a pivotal role in modeling and resolving complex spatial challenges in various scientific and engineering fields

### Uses in quantifying spatial relationships

3D vectors are used to quantify the size, direction, and orientation of objects in three-dimensional space

Algorino

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