Vectors

Vectors, in mathematical terms, are entities that possess both direction and magnitude (size). This differentiates them from scalars, which only contain magnitude. In the field of physics, vectors are employed to illustrate quantities such as force and velocity. In the realm of computer science, they are utilized for spatial representations and algorithms. Operations such as addition, subtraction, and multiplication by scalars can be performed on vectors.

Finding the Position Vector

Given two vectors \(\mathbf{A} = 3\mathbf{i} + 2\mathbf{j}\) and \(\mathbf{B} = \mathbf{i} + 4\mathbf{j}\), find the position vector \(\mathbf{R}\) such that \(\mathbf{R} = 2\mathbf{A} - \mathbf{B}\).

Finding an Orthonormal Basis by Gram-Schmidt Method

Given vectors \( \vec{a} = (1, 0, 0) \), \( \vec{b} = (1, 1, 0) \), and \( \vec{c} = (1, 1, 1) \), find an orthonormal basis by using the Gram-Schmidt Method.