Lesson 11: Mathematics: Coordinate Geometry, Vectors and 3D
Coordinate geometry turns pictures into algebra: a line is an equation, a circle is a completed square, and a vector is a directed quantity you can add, dot and cross. Three-dimensional geometry extends the same tools with one more coordinate and one more direction to keep track of.
This lesson covers straight lines, circles, vectors in two and three dimensions, and lines and planes in space, with the distance formulas that appear most often.
Lines, circles and vectors
The slope of a line through (x₁, y₁) and (x₂, y₂) is (y₂ − y₁)/(x₂ − x₁). A circle's equation is completed into (x − h)² + (y − k)² = r², giving centre (h, k) and radius r. The dot product a·b = a₁b₁ + a₂b₂ + a₃b₃ is zero exactly when the vectors are perpendicular.
Slope through (2, 1) and (5, 7): (7 − 1)/(5 − 2) = 6/3 = 2.
Circle x² + y² − 4x + 6y − 12 = 0 completes to (x − 2)² + (y + 3)² = 12 + 4 + 9 = 25, so centre (2, −3) and radius 5.
Dot product (1, 2, 3)·(4, −1, 2) = 4 − 2 + 6 = 8, which is non-zero, so the vectors are not perpendicular.
Three-dimensional geometry
The distance between (x₁, y₁, z₁) and (x₂, y₂, z₂) is √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]. The distance from a point to a plane ax + by + cz + d = 0 is |ax₀ + by₀ + cz₀ + d| / √(a² + b² + c²). The cross product's magnitude is the area of the parallelogram spanned by the two vectors.
Distance between (1, 2, 3) and (4, 6, 3): √[(3)² + (4)² + 0] = √25 = 5.
Distance from (1, 2, 3) to the plane 2x + y − 2z + 3 = 0: |2 + 2 − 6 + 3| / √(4 + 1 + 4) = |1|/3 = 1/3.
Vectors (1, 2, 2) and (2, 1, −2) have dot product 2 + 2 − 4 = 0, so they are perpendicular (angle 90°).
A triangle spanned by vectors (3, 0, 0) and (0, 4, 0) has cross-product magnitude 12, so its area is ½ × 12 = 6.
Common pitfalls
Do not forget the square root in a distance formula, or the absolute value in a plane distance: both are needed for a length, which is never negative.
Do not read a circle's centre from the uncompleted equation: complete the square first, or the signs of h and k will be reversed.
Further study
- NCERT Class 11 Mathematics (straight lines, conic sections, introduction to three-dimensional geometry) and Class 12 Mathematics (vectors, three-dimensional geometry).
📝 Quiz
1. What is the slope of the line through (2, 1) and (5, 7)?
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b) (7 − 1)/(5 − 2) = 6/3 = 2.
2. What are the centre and radius of x² + y² − 4x + 6y − 12 = 0?
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a) Completing the square gives (x − 2)² + (y + 3)² = 25.
3. What is (1, 2, 3)·(4, −1, 2)?
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b) 4 − 2 + 6 = 8.
4. What is the distance between (1, 2, 3) and (4, 6, 3)?
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b) √(9 + 16 + 0) = √25 = 5.