Position Vector and Components of a Vector

Position Vector of a Point

Fix an origin $O$. For any point $P(x,y,z)$ in space, the vector $\vec{OP}$ is called the position vector of $P$ relative to $O$:

Position vectors let every point in space be treated as a vector from a common origin, which is what makes vector methods (rather than pure coordinate geometry) so effective for problems about lines, planes and geometric figures.

Components of a Vector

Any vector $\vec a$ can be resolved along the three coordinate axes into components $a_1,a_2,a_3$ so that

Here $a_1,a_2,a_3$ are scalars, called the scalar components of $\vec a$; the vectors $a_1\hat i,a_2\hat j,a_3\hat k$ are its vector components. The magnitude of $\vec a$ in terms of its components is

The Vector Joining Two Points

If $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ are any two points, with position vectors $\vec{OA}=x_1\hat i+y_1\hat j+z_1\hat k$ and $\vec{OB}=x_2\hat i+y_2\hat j+z_2\hat k$, then by the triangle law of addition (Section 5), $\vec{OA}+\vec{AB}=\vec{OB}$, so

This is the single most-used formula in the chapter: it converts any problem about two labelled points into a problem about a component vector, and the same subtraction pattern ("terminal point minus initial point") reappears in the section formula, the dot product and the cross product.

Finding a Unit Vector Along a Given Direction …