# Understanding components of a vector

I learned that we can get the component of a vector in any direction using the dot product. The problem I have is the meaning of the term component itself. The component of a vector $\vec A$ in the direction of the unit vector $\vec u$ is $$|\vec A||\vec u|\cos (\theta).$$

1) Will the other component be $|\vec A|\cos (90-\theta)$?

2) Do any two components must be perpendicular or can they be any two vectors whose sum will be vector $\vec A$?

3) If the angle $\theta$ between vector $\vec A$ and the unit vector that indicates the direction of the component I want is $90 <\theta<180$

The component will have a negative sign

Does this sign mean that no component in the direction of the unit vector and that the only component will be in the opposite direction?

• Have you encountered systems of linear differential equations and or linear transformations? These can give you examples of coordinate systems not spanned by orthonormal vectors – Triatticus Jun 16 '16 at 2:03

## 1 Answer

$\DeclareMathOperator{\proj}{proj}$The term "component" has several different (but partially-overlapping) meanings in linear algebra. These include:

• Your definition, which is a scalar: If $u$ is a unit vector, then the component of $A$ on $u$ is the scalar $c$ such that $$cu = (|A| \cos\theta) u = (A \cdot u) u = \proj_{u}A$$ is the orthogonal projection of $A$ on $u$.

• Some people call the projection itself, which is a vector, the "component" of $A$ in the direction of $u$.

• If $(e_{1}, e_{2}, \dots, e_{n})$ is a basis for a vector space, and if $$A = A_{1} e_{1} + A_{2} e_{2} + \dots + A_{n} e_{n} = \sum_{j=1}^{n} A_{j} e_{j}, \tag{1}$$ the scalars $(A_{1}, A_{2}, \dots, A_{n})$ are sometimes called the "components" of $A$ with respect to the basis $(e_{j})_{j=1}^{n}$.

The first and second definitions are clearly related. The first and third are closely related for an orthonormal basis: If the $(e_{j})$ are mutually-orthogonal unit vectors, then $$A_{i} = A \cdot e_{i}.$$ (Proof: Dot both sides of (1) with $e_{i}$; orthonormality says $e_{i} \cdot e_{j} = 0$ if $i \neq j$.) Moreover, the vectors $A_{i} e_{i}$ are mutually-orthogonal.

As for your numbered questions:

1. "Components of $A$" make sense only relative to some vector or basis. That is, there's no notion of "other component" unless $A$ is a plane vector and $u$ is one vector of an orthonormal basis.

2. As defined, components aren't vectors at all. If you consider components to be vectors, it's common to assume you're working in an orthonormal basis. If that's the case (which is common in physics and engineering), then "yes", components must be perpendicular.

3. Briefly, "yes": If $A$ and $u$ make an angle $\theta$ greater than a right angle, then $\cos\theta < 0$. Geometrically, you can interpret this by noting that $A$ and $-u$ make an angle smaller than a right angle, and the component of $A$ on $-u$ is the negative of the component of $A$ on $u$.