# 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 Jun 16, 2016 at 2:03

$\DeclareMathOperator{\proj}{proj}$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.

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.
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 $u$.