Straight line and the plane – the basic concepts of geometry. These two-dimensional and three-dimensional figures which are bases for creation of any flat and spatial designs. Always it is possible to calculate a corner between a straight line and the plane on their equations.
Instruction
1. A straight line and the plane – two interconnected geometrical concepts. Through any two points of the plane it is possible to draw the straight line consisting of its points. And any straight line belongs to any plane. Any figure in geometry – set of the crossed lines and sites of a surface limited to them, from protozoa of a triangle and circle to non-standard convex polygons and prisms.
2. For each straight line it is possible to find a projection to some plane in space. Thus, it is possible to calculate a corner between them as adjacent to the corner formed by vectors of the direction and a normal. For example, let the initial equation of a straight line of L and the general equation of plane P is set: L: (x – h0) / p = (at – u0) / r = (z – z0) / s; P: And • x + B • at + C·z + D = 0.
3. Coefficients of these equations are coordinates of a vector of the direction of a straight line and a vector of a normal for the plane. Then the problem of definition of a corner between a straight line and its projection comes down to search of an adjacent corner between these vectors. The adjacent corner in this situation makes in the sum with required 90 ° or π/2. Find a cosine of the angle (π/2 – α) on the known formula: cos (π/2 – α) = sin α = | p • And + r · B + s•C | / (√ (p² + r² + s²) • √ (And² + B² + C²)).
4. Special cases when this corner is equal 90 ° or 180 °, are the proof of their perpendicularity or parallelism. Then: • if And / to p = to B/r = With / s – the straight line is perpendicular to the plane; • if A·p + to B · r + C·s = 0 – the straight line is parallel to the plane.
5. Example: to find a corner between a straight line (x - 1)/4 = (at + 3)/-2 = (z - 8)/1 and plane 5 • x + 3 • at – 4•z = 0. ReshenieVypishite of the coordinate of a vector of the direction of a straight line – (4,-2, 1) and a normal vector of the plane – (5, 3,-4). Substitute all values in a formula of a sine of the angle: sin α = | 20 – 6 - 4 | / (√ (16 + 4 + 1) • √ (25 + 9 + 16)) ≈ 0.3.
6. Calculate an arcsine of the turned-out size to determine a required corner α:α = by arcsin 0.3 ≈ 17.46 °.