Vectores en 3D: Rectangular, Unitarios, Esféricos y Cilíndricos

Vectors in Three Dimensions

Introduction to 3D Vectors

• The video begins with an introduction to vectors in three dimensions, highlighting the concept of a vector represented as a ternary of coordinates (x, y, z).

• A vector in its rectangular form now includes three components: x, y, and z. Each component corresponds to a unit vector notation: hati, hatj, hatk .

Calculating the Magnitude of a Vector

• The magnitude of a 3D vector follows the Pythagorean theorem: |V| = sqrtx^2 + y^2 + z^2 . For example, for the vector (3, 4, 5), it calculates as |V| = sqrt3^2 + 4^2 + 5^2 = 7.071 .

• Unlike two-dimensional vectors, three-dimensional vectors do not have a polar representation due to multiple reference planes; instead, they can be expressed in spherical coordinates.

Spherical Representation of Vectors

• In spherical coordinates for a 3D vector, two angles are defined: the polar angle (θ) between the Z-axis and the vector's magnitude and the azimuthal angle (φ) between the projection on the XY-plane and the X-axis.

• The conversion from Cartesian to spherical involves calculating these angles based on their respective formulas.

Calculating Polar Angle

• The polar angle is calculated using cos(θ) = z/r , where r is the magnitude. For instance, with r being 5sqrt2 , we find that θ equals cos^-1(1/sqrt2) = 45° .

Calculating Azimuthal Angle

• The azimuthal angle φ is determined by cos(φ) = x/sqrtx^2 + y^2 . Using x = 3 and y = 4 leads to φ being approximately 53.13°.

• Thus, when expressing our original vector in spherical form it becomes: module r = 5sqrt2 , polar angle θ at 45°, and azimuthal angle φ at approximately 53.13°.

Summary of Vector Forms

Understanding Cylindrical Coordinates

Introduction to Cylindrical Coordinates

• The discussion begins with the concept of cylindrical coordinates, emphasizing that they are represented by three parameters: rho , the azimuthal angle, and z .

• The azimuthal angle is defined as the angle between the x-axis and y-axis in a two-dimensional plane, which is crucial for understanding how vectors are positioned in space.

Calculating Parameters

• An example vector (3, 4, 5) is used to illustrate how to calculate the azimuthal angle in cylindrical coordinates. The calculated angle is approximately 53.13 degrees.

• The radius rho of the cylinder is derived using the Pythagorean theorem: sqrtx^2 + y^2 . For the vector (3, 4), this results in a radius of 5.

Final Representation in Cylindrical Coordinates

• The final representation of the vector in cylindrical coordinates includes R = 5 , an azimuthal angle of 53.13 degrees, and a height z = 5 .

• In three-dimensional space, an additional coordinate (z-axis) and unit vector (k-direction) are introduced. This allows for modeling vectors not just as points but also as entities capable of rotation within three dimensions.

Summary Insights

• Understanding cylindrical coordinates involves translating rectangular coordinates into a format that incorporates angles and radii.

• Key calculations include determining both radius and angles using basic geometric principles like Pythagoras' theorem.