Vectors of different dimensions cannot be added or subtracted directly because their structures are incompatible. Each component in one vector must correspond to a component in the other. For example, you cannot add [2, 3] and [1, 4, 5] because the first has two components and the second has three. Attempting to do so will result in a dimension mismatch error in most programming languages. Scalar multiplication, however, does not require dimensional compatibility because it affects each component of a single vector individually. So you can multiply a 2D, 3D, or even higher-dimensional vector by a scalar without issue. The scalar is applied uniformly across all components regardless of the number of dimensions. This rule helps ensure the mathematical consistency of vector spaces and prevents incorrect computations. In summary, addition and subtraction require equal dimensions, but scalar multiplication can be applied to any vector as long as all components are numeric.

Yes, vector operations can be chained together in one expression, and they follow the rules of standard arithmetic order of operations. Scalar multiplication is treated like multiplication and is evaluated before addition unless parentheses indicate otherwise. For example, in the expression 2u + 3v, you first multiply vector u by 2 and vector v by 3, then add the resulting vectors together. So if u = [1, 2] and v = [3, 4], you calculate 2u = [2, 4] and 3v = [9, 12], then add them: [2, 4] + [9, 12] = [11, 16]. Parentheses can be used to override this order. For example, if you have 2(u + v), you must first add u and v, then multiply the result by 2. This flexibility allows complex transformations to be performed in a compact form and is especially useful in programming environments, physics simulations, and computer graphics where such operations occur frequently.

Negative vectors are simply vectors that point in the exact opposite direction to their original counterparts. You obtain a negative vector by multiplying all its components by -1. For example, the negative of vector [2, 3] is [-2, -3]. This concept is directly related to vector subtraction. Subtracting one vector from another, such as u - v, is equivalent to adding u and the negative of v: u + (-v). So if u = [4, 5] and v = [1, 2], then u - v = [4, 5] + [-1, -2] = [3, 3]. This interpretation allows vector subtraction to be understood as combining one movement or direction with the exact opposite of another. In geometric terms, subtracting v from u produces a new vector that points from the tip of v to the tip of u, assuming both start at the origin. This helps in understanding relative position, direction, and displacement in vector-based systems.

The zero vector is a vector where all components are zero, such as [0, 0] in 2D or [0, 0, 0] in 3D. It has no direction and its magnitude is zero. In vector operations, the zero vector acts as the additive identity, meaning any vector added to the zero vector remains unchanged: u + 0 = u. When used in scalar multiplication, any vector multiplied by 0 becomes the zero vector, effectively removing its magnitude and direction. For example, 0 × [3, 4] = [0, 0]. In programming and computational mathematics, the zero vector can serve as a default value or as an indicator that no movement or change has occurred. It's also used in linear algebra to represent the result of a system in equilibrium. Visually, the zero vector is a point at the origin with no length, and it cannot be normalised since normalisation involves dividing by its magnitude, which is zero.

Vector operations behave in the same fundamental way regardless of the number of dimensions. In 4D, 5D, or even higher-dimensional spaces, vector addition still involves adding corresponding components, and scalar multiplication still involves multiplying each component by a scalar. For example, in 4D:
u = [1, 2, 3, 4]
v = [4, 3, 2, 1]
u + v = [5, 5, 5, 5]
2u = [2, 4, 6, 8]
These operations follow the same rules and properties described in lower dimensions. However, visualising these operations becomes much harder beyond 3D, since humans cannot perceive higher spatial dimensions. Instead, such vectors are often used in data science, machine learning, and computer science to represent feature sets, transformations, and abstract mathematical spaces. The mathematics remains consistent, and vector operations still model translations, scalings, and combinations of values in these abstract spaces just as they do in 2D or 3D.