Vectors Joining Two Points

You might already have come across the term vector. A quantity which has magnitude, as well as direction, is represented by a vector. The most common example of a vector in mathematics is a directed line segment. Let us say that a line AB, 5 cm in length, is pointed towards the south. In this case, AB is a vector. There are different types of vectors. Basic mathematical operations can be applied to them as well. In this article, we will talk about a vector which joins two points.

Vectors Joining Two Point

We can represent a point by its x coordinate, y coordinate and z coordinate. Let us say there are two points represented by their x-, y- and z- coordinates as:

  • P1(x1,y1,z1)
  • P2(x2,y2,z2)

We join the points P1 and P2 by a vector and call it as P1P2.

Vectors Joining Two Points

We represent the vectors from the origin O along the x-, y- and z-axes as i, j and k respectively. Now we join the origin O to P1 with the vector OPand to P2 with the vector OP2. Using the triangle law, we get:

\(\overrightarrow{OP_{1}}+ \overrightarrow{P_{1}P_{2}} = \overrightarrow{OP_{2}}\)


\(\overrightarrow{P_{1}P_{2}} = \overrightarrow{OP_{2}}- \overrightarrow{OP_{1}}\)

That is,

\(\overrightarrow{P_{1}P_{2}} = (x_{2}\hat{i} + y_{2}\hat{j}+ z_{2}\hat{k}) – (x_{1}\hat{i} + y_{1}\hat{j}+ z_{1}\hat{k})\)

\(= (x_{2} – x_{1})\hat{i} + (y_{2} – y_{1} ) \hat{j}+ (z_{2}- z_{1})\hat{k}\)

Thus, the above equation represents the vector P1P2. Its magnitude can be given by:

\(\overrightarrow{P_{1}P_{2}} = \sqrt{(x_{2}- x_{1})^{2} + (y_{2}- y_{1})^{2} + (z_{2}- z_{1})^{2}}\)


Question: Find the vector and its magnitude which joins the point A with coordinates (4, 5, 6) to point B with coordinates (10, 11, 12).

Solution: The vector is directed from the point A to B and can be denoted by \(\overrightarrow{AB}\)


\(\overrightarrow{AB} = (10 – 4) \hat{i} + (11 – 5) \hat{j} + (12 – 6) \hat{k} = 6 \hat{i} + 6 \hat{j} + 6 \hat{k}\)

Magnitude can be given by:

\(\overrightarrow{AB} = \sqrt{6^{2} + 6^{2} + 6^{2}} = \sqrt{108} = 10.39\)<

To learn more about vectors, section formula and other vector concepts with illustrative examples, by watching interactive video lectures on them, download Byju’s The Learning App.

Practise This Question

Which point is interior to ABC?