Definition:Vector Quantity/Component/Cartesian 3-Space

Definition
Let $\mathbf a$ be a vector quantity embedded in Cartesian $3$-space $S$.

Let $\mathbf a$ be represented with its initial point at the origin of $S$.

Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively.

Then:
 * $\mathbf a = x \mathbf i + y \mathbf j + z \mathbf k$

where:
 * $x \mathbf i$, $y \mathbf j$ and $z \mathbf k$ are the component vectors of $\mathbf a$ in the $\mathbf i, \mathbf j, \mathbf k$ directions
 * $x$, $y$ and $z$ are the components of $\mathbf a$ in the $\mathbf i$, $\mathbf j$ and $\mathbf k$ directions.

It is usual to arrange that the coordinate axes form a right-handed Cartesian $3$-space.

It is usually more convenient to write $\mathbf a$ as the ordered tuple $\tuple {x, y, z}$ instead of $\mathbf a = x \mathbf i + y \mathbf j + z \mathbf k$.

Also see

 * Definition:Component of Vector in Plane