Edexcel IAL - Pure Maths 4- 7.5 Distance Between Two Points- Study notes - New syllabus
Edexcel IAL – Pure Maths 4- 7.5 Distance Between Two Points -Study notes- New syllabus
Edexcel IAL – Pure Maths 4- 7.5 Distance Between Two Points -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- 7.5 Distance Between Two Points
The Distance Between Two Points
The distance between two points in space is the length of the straight line joining them.
If two points are:
\( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \)
then the distance \( d \) between them is given by:
\( d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2} \)
This formula is obtained from Pythagoras’ theorem in three dimensions.
Special Case: Two Dimensions
If \( z_1 = z_2 = 0 \), the formula becomes:
\( d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} \)
Example
Find the distance between \( A(1,2) \) and \( B(4,6) \).
▶️ Answer / Explanation
\( d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9+16} = 5 \)
Example
Find the distance between \( P(-2,3,1) \) and \( Q(4,-1,5) \).
▶️ Answer / Explanation
\( d = \sqrt{(4+2)^2 + (-1-3)^2 + (5-1)^2} \)
\( d = \sqrt{36 + 16 + 16} = \sqrt{68} = 2\sqrt{17} \)
Example
Points \( A(1,2,-1) \) and \( B(k,5,3) \) are 6 units apart. Find the value of \( k \).
▶️ Answer / Explanation
\( 6^2 = (k-1)^2 + (5-2)^2 + (3+1)^2 \)
\( 36 = (k-1)^2 + 9 + 16 \)
\( (k-1)^2 = 11 \)
\( k = 1 \pm \sqrt{11} \)