Three-dimensional geometry — KCSE Mathematics

In three-dimensional geometry, the distance between two points can be calculated using the distance formula. Given two points, A(x₁, y₁, z₁) and B(x₂, y₂, z₂), the distance d between them is given by the formula:

d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²).

Steps to Calculate Distance:

1. Identify the coordinates of the two points.
2. Substitute the coordinates into the distance formula.
3. Simplify the expression under the square root.
4. Calculate the square root to find the distance.

Example:

Calculate the distance between points A(1, 2, 3) and B(4, 6, 8).

1. Identify coordinates: A(1, 2, 3), B(4, 6, 8).
2. Substitute: d = √((4 - 1)² + (6 - 2)² + (8 - 3)²).
3. Simplify: d = √(3² + 4² + 5²) = √(9 + 16 + 25).
4. Calculate: d = √50 = 5√2.

Thus, the distance between points A and B is 5√2 units.

In three-dimensional geometry, calculating angles is essential for understanding spatial relationships. To find the angle between two lines, use the direction ratios of the lines. For a line and a plane, apply the formula:

Angle (θ) = cos⁻¹((d1 · n) / (|d1| |n|))
where:

• d1 is the direction ratio of the line
• n is the normal vector of the plane
• · denotes the dot product
• | | denotes magnitude

To find the angle between two planes, use their normal vectors. The formula is similar:

Angle (φ) = cos⁻¹((n1 · n2) / (|n1| |n2|))
where n1 and n2 are the normal vectors of the planes.

It is important to ensure that all vectors are defined correctly and that you calculate the dot product accurately.