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Position Vector — Definition, Formula & Examples

A position vector is an arrow that starts at the origin and ends at a specific point. It tells you exactly where that point is, both its direction from the origin and its distance from it.

For a point PP with coordinates (x,y)(x, y) in R2\mathbb{R}^2 (or (x,y,z)(x, y, z) in R3\mathbb{R}^3), the position vector of PP is r⃗=OP→\vec{r} = \overrightarrow{OP}, where OO is the origin. Its components equal the coordinates of PP: r⃗=⟨x,y⟩=x i+y j\vec{r} = \langle x, y \rangle = x\,\mathbf{i} + y\,\mathbf{j}. A position vector is tied to the origin as its starting point. A general vector can start anywhere.

Key Formula

r⃗=OP→=⟨x,y⟩,AB→=b⃗−a⃗=⟨b1−a1,  b2−a2⟩\vec{r} = \overrightarrow{OP} = \langle x, y \rangle, \qquad \overrightarrow{AB} = \vec{b} - \vec{a} = \langle b_1 - a_1,\; b_2 - a_2 \rangle
Where:
  • r⃗\vec{r} = Position vector of point P
  • OO = The origin, (0, 0)
  • (x,y)(x, y) = Coordinates of point P
  • a⃗=⟨a1,a2⟩\vec{a} = \langle a_1, a_2 \rangle = Position vector of point A
  • b⃗=⟨b1,b2⟩\vec{b} = \langle b_1, b_2 \rangle = Position vector of point B
  • AB→\overrightarrow{AB} = Vector from A to B (displacement)

Worked Example

Problem: Points A(2,1)A(2, 1) and B(5,5)B(5, 5) lie in the plane. Write their position vectors, then use them to find AB→\overrightarrow{AB} and its length.
Step 1: Read each position vector directly from the point's coordinates.
a⃗=⟨2,1⟩,b⃗=⟨5,5⟩\vec{a} = \langle 2, 1 \rangle, \qquad \vec{b} = \langle 5, 5 \rangle
Step 2: Subtract the starting point's position vector from the ending point's position vector.
AB→=b⃗−a⃗=⟨5−2,  5−1⟩=⟨3,4⟩\overrightarrow{AB} = \vec{b} - \vec{a} = \langle 5 - 2,\; 5 - 1 \rangle = \langle 3, 4 \rangle
Step 3: Find the length of AB→\overrightarrow{AB} using the magnitude formula.
∣AB→∣=32+42=25=5\left|\overrightarrow{AB}\right| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5
Answer: a⃗=⟨2,1⟩\vec{a} = \langle 2, 1 \rangle, b⃗=⟨5,5⟩\vec{b} = \langle 5, 5 \rangle, AB→=⟨3,4⟩\overrightarrow{AB} = \langle 3, 4 \rangle, and the distance from A to B is 5.

Why It Matters

In AP Calculus BC and AP Physics C, motion is described by a position vector r⃗(t)=⟨x(t),y(t)⟩\vec{r}(t) = \langle x(t), y(t) \rangle that changes with time. Its derivative r⃗ ′(t)\vec{r}\,'(t) is the velocity vector, and the derivative of velocity is acceleration. Game engines and robotics software also store every object's location as a position vector.

Common Mistakes

Mistake: Computing AB→\overrightarrow{AB} as a⃗−b⃗\vec{a} - \vec{b}, which gives ⟨−3,−4⟩\langle -3, -4 \rangle in the example above.
Correction: Use end minus start: AB→=b⃗−a⃗\overrightarrow{AB} = \vec{b} - \vec{a}. Reversing the order gives BA→\overrightarrow{BA}. That vector has the same length but points the opposite way.

Related Terms