The magnitude of a vector can be calculated by taking the square root of the sum of the squared components of the vector. In other words, if a vector is represented by a = (a1, a2, a3), then the magnitude is given by √(a12 + a22 + a32).
To add or subtract vectors, you simply add or subtract the corresponding components of each vector. For example, if you have two vectors a = (a1, a2, a3) and b = (b1, b2, b3), then a + b = (a1 + b1, a2 + b2, a3 + b3).
The dot product of two vectors is calculated by multiplying the corresponding components of each vector and then adding the products together. In other words, if a = (a1, a2, a3) and b = (b1, b2, b3), then a · b = a1b1 + a2b2 + a3b3.
The angle between two vectors can be calculated using the dot product formula. If a and b are two vectors, then the angle between them (in radians) can be found using the equation cosθ = a · b / (∥a∥ ∥b∥), where ∥a∥ and ∥b∥ represent the magnitudes of the vectors.
To multiply a vector by a scalar (a single number), you simply multiply each component of the vector by that number. For example, if a = (a1, a2, a3) and k is a scalar, then k · a = (k * a1, k * a2, k * a3).