Edexcel IAL - Pure Maths 4- 7.4 Position Vectors- Study notes  - New syllabus

Position Vectors

The position vector of a point is the vector that joins the origin \( O(0,0,0) \) to the point.

If a point \( A \) has coordinates \( (x,y,z) \), then its position vector is written as:

\( \vec{OA} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \)

This tells both the direction and distance of the point from the origin.

Position Vectors in Two Dimensions

If \( A(x,y) \), then:

\( \vec{OA} = x\mathbf{i} + y\mathbf{j} \)

Vector Between Two Points

If:

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

then:

\( \vec{AB} = \vec{OB} – \vec{OA} \)

\( \vec{AB} = (x_2-x_1)\mathbf{i} + (y_2-y_1)\mathbf{j} + (z_2-z_1)\mathbf{k} \)

Geometrical Meaning

• Position vectors locate points in space.
• Subtracting position vectors gives displacement.
• They can be used to describe lines and shapes.