The magnitude of a vector is calculated by taking the square root of the sum of the squares of its components. This can be represented mathematically as √(x² + y² + z²), where x, y, and z are the components of the vector in the x, y, and z directions, respectively.
Unit vectors are used to represent direction and orientation in a vector space. They have a magnitude of 1 and are used to scale other vectors in order to find their components in a given direction.
The dot product of two vectors is calculated by multiplying their corresponding components and then summing the results. This can be represented mathematically as a·b = ax * bx + ay * by + az * bz, where a and b are the two vectors and ax, ay, az and bx, by, bz are their components in the x, y, and z directions, respectively.
A unit vector has a magnitude of 1, while a regular vector can have any magnitude. Unit vectors are used to represent direction and are often used in calculations involving other vectors.
The angle between two vectors can be found by using the dot product formula and the magnitude formula. The angle between two vectors a and b can be represented as θ = cos⁻¹((a·b)/(|a||b|)), where |a| and |b| are the magnitudes of the two vectors.