Examples of Vectors Joining Two Points

Example 1: Given the points P(1, 2, 3) and Q(4, 5, 6) now find this

1. Find the vector PQ that joins point P to point Q.
2. Calculate the magnitude of vector PQ.

Solution:

Let us consider two point P(x_1,y₁,z₁) and Q(x₂,y₂,z₂) in 3d space. After comparison we get the value of x₁ = 1, y₁ = 2, z₁ = 3, x₂ = 4, y₂ = 5, z₂ = 6

1. We know that the vectors joining two points P and Q is
[Tex]\overrightarrow{PQ}= (x_2-x_1)\hat{\imath} + (y_2-y_1)\hat{\jmath} + (z_2-z_1)\hat{k}[/Tex]
[Tex]\overrightarrow{PQ}= (4-1)\hat{\imath} + (5-2)\hat{\jmath} + (6-3)\hat{k}[/Tex]
[Tex]\overrightarrow{PQ}= 3\hat{\imath} + 3\hat{\jmath} + 3\hat{k}[/Tex]

2. We know that magnitude of the vector PQ is
[Tex]|PQ| =\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}[/Tex]
[Tex]|PQ| =\sqrt{(4-1)^2 + (5-2)^2 + (6-3)^2}[/Tex]
|[Tex]PQ| =\sqrt{27}[/Tex]
[Tex]|PQ| =3\sqrt{3}[/Tex]

Example 2: Given the points P(1, 2) and Q(4, 5) now find this

1. Find the vector PQ that joins point P to point Q.
2. Calculate the magnitude of the vector PQ.

Solution:

1. We know that the vectors joining two points P and Q is
[Tex]\overrightarrow{PQ}= (x_2-x_1)\hat{\imath} + (y_2-y_1)\hat{\jmath} + (z_2-z_1)\hat{k}[/Tex]
[Tex]\overrightarrow{PQ}= (4-1)\hat{\imath} + (5-2)\hat{\jmath} + (0-0)\hat{k}[/Tex]
[Tex]\overrightarrow{PQ}= 3\hat{\imath} + 3\hat{\jmath} +0\hat{k}[/Tex]

2. We know that magnitude of the vector PQ is
[Tex]|PQ| =\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}[/Tex]
[Tex]|PQ| =\sqrt{(4-1)^2 + (5-2)^2 + (0-0)^2}[/Tex]
[Tex]|PQ| =\sqrt{(3)^2 + (3)^2 + (0)^2}[/Tex]
[Tex]|PQ| =\sqrt{18}[/Tex]
[Tex]|PQ| =3\sqrt{2}[/Tex]

Example 3: Find the vector joining the points P(1, 0,0) and Q(0, 1, 0) directed to Q from P.

Solution:

Vector which is directed from the point P to Q can be written as
[Tex]\begin{array}{l}\vec{PQ}\end{array}[/Tex]
Therefore,,
[Tex]\begin{array}{l}\vec{PQ}=(0-1)\hat{i}+(1-0)\hat{j}+(0-0)\hat{k}\\=-1\hat{i}+1\hat{j}+0\hat{k}\end{array}[/Tex]

Vector quantities are the quantities that have both magnitude and direction. Vectors are key in physics, engineering, and computer science for describing magnitude and direction. Vectors can be easily represented in 2-D or 3-D spaces. Connecting two points with vectors is essential for understanding advanced topics like vector algebra and motion. They are subject to basic mathematical operations much like other mathematical elements.

In this article, we will understand about the Vectors Joining Two Points.