Vector Joining Two Points in Algebra

In three-dimensional coordinate geometry, the vector joining two points tells us the displacement from one point (initial point) to another (terminal point) and is fundamental for distance, section formula, and many 3D geometry problems

If \[P_1(x_1, y_1, z_1)\] and \[P_2(x_2, y_2, z_2)\] are two points in space, then the vector joining \[P_1\] to \[P_2\] is the vector 

\[\vec{P_1P_2}\]

representing the displacement from \[P_1\] (initial point) to \[P_2\] (terminal point).

Magnitude of vector: 

Find the vector joining P(2, 3, 0) and Q(-1, -2, -4) directed from P to Q.

Solution:

• Initial point: P(2, 3, 0)

• Terminal point: Q(-1, -2, -4)

Magnitude

• Initial point: starting point of vector; terminal point: ending point.

• Vector joining \[P_1(x_1, y_1, z_1)\] to \[P_2(x_2, y_2, z_2)\]:

  \[\vec{P_1P_2} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}\]

• Order matters: \[\vec{P_1P_2} = -\vec{P_2P_1}\]

• Magnitude equals distance between points:

  \[|\vec{P_1P_2}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]