Position Vector

A position vector locates a point in space by describing its distance and direction from the origin. Instead of giving coordinates directly, the position vector tells how far to move along each axis starting from the origin.

If a point \(P(x, y)\) lies in the plane, the position vector of P is the vector drawn from the origin O to the point P.

It is written as:

\[ \vec{OP} = \langle x, y \rangle \]

In three dimensions, for a point \(P(x, y, z)\), the position vector is:

\[ \vec{OP} = \langle x, y, z \rangle \]

The coordinates of a point and its position vector contain the same information. The difference is only in how we think about them:

• Coordinates tell where the point lies.
• The position vector tells how to reach the point from the origin.

This makes position vectors useful in geometry and physics, where direction and magnitude matter.

For a point in space, the position vector includes all three components.

If \(P(2, -1, 5)\), then:

\[ \vec{OP} = \langle 2, -1, 5 \rangle \]

Its magnitude is:

\[ |\vec{OP}| = \sqrt{2^2 + (-1)^2 + 5^2} = \sqrt{30} \]

The position vector also helps find the vector from one point to another. If two points are:

\[ A(x_1, y_1, z_1), \; B(x_2, y_2, z_2) \]

Then the vector \(\overrightarrow{AB}\) is:

\[ \overrightarrow{AB} = \langle x_2 - x_1,\; y_2 - y_1,\; z_2 - z_1 \rangle \]

This is formed by subtracting the position vector of A from the position vector of B.

Position vectors make geometric ideas easier to express. For example:

• The midpoint of AB can be written using position vectors.
• The direction of a line can be given by subtracting position vectors.
• Distances can be computed as magnitudes of difference vectors.