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.
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.
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 OP1 and to P2 with the vector OP2. Using the triangle law, we get:
\(\overrightarrow{OP_{1}}+ \overrightarrow{P_{1}P_{2}} = \overrightarrow{OP_{2}}\)
Or
\(\overrightarrow{P_{1}P_{2}} = \overrightarrow{OP_{2}}- \overrightarrow{OP_{1}}\)
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:
\(\overrightarrow{P_{1}P_{2}} = \sqrt{(x_{2}- x_{1})^{2} + (y_{2}- y_{1})^{2} + (z_{2}- z_{1})^{2}}\)
Example-
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 \(\overrightarrow{AB}\)
Thus,
\(\overrightarrow{AB} = (10 – 4) \hat{i} + (11 – 5) \hat{j} + (12 – 6) \hat{k} = 6 \hat{i} + 6 \hat{j} + 6 \hat{k}\)
Magnitude can be given by:
\(\overrightarrow{AB} = \sqrt{6^{2} + 6^{2} + 6^{2}} = \sqrt{108} = 10.39\)