Vector Components and Scalar Components
In math and physics we often have to deal with vectors in component form. \(\vec{u} = 2\hat{i} - 3\hat{j}\) This is a vector in component form. \(2\hat{i}\) and \(-3\hat{j}\) are its vector components , while the multiples of \(\hat{i}\) and \(\vec{j}\), i.e., 2 & -3, are its scalar components . Expressing a vector in component form basically means writing that vector in terms of some “base” vectors like \(\hat{i}\) and \(\hat{j}\). \(\hat{i}\) is a vector of length one unit in the positive x-direction; \(\hat{j}\) is a vector of length one unit in the positive y-direction. Let’s say vector u’s tail is at the origin. Starting from the origin, we travel two times the length of \(\hat{i}\)\((2 \times 1 = 2)\) in the positive x-direction, and then travel 3 times the length of \(\hat{j}\)\((3 \times 1 = 3)\) in the negative y-direction(because the vector component is negative). We have reached the head of the vector u. So when a vector's tail is at the origin...