The straight line in space is set by the initial equation, containing coordinates of its guides of vectors. Proceeding from it, it is possible to determine a corner between straight lines by a formula of the cosine of the angle formed by vectors.
1. It is possible to define a corner between two straight lines in space even if they are not crossed. In this case it is necessary to combine mentally the beginnings of their guides of vectors and to calculate the size of the turned-out corner. In other words, it is any of the adjacent corners formed by the skew lines drawn parallel to data.
2. There are several tasksways straight line in space, for example, vector and parametrical, parametrical and initial. It is convenient to use three mentioned methods when finding a corner since all of them assume introduction of coordinates of the directing vectors. Knowing these sizes, it is possible to determine an educated corner by the theorem of cosines from vector algebra.
3. Let's assume, two straight lines of L1 and L2 are set by the initial equations: L1: (x – x1) / k1 = (y – y1) / l1 = (z – z1) / n1; L2: (x – x2) / k2 = (y – y2) / l2 = (z – z2) / n2.
4. Using the sizes ki, li and ni, write down coordinates of the directing vectors of straight lines. Call them N1 and N2: N1 = (k1, l1, n1); N2 = (k2, l2, n2).
5. The formula for a cosine of the angle between vectors represents a ratio between their scalar product and result of arithmetic multiplication of their lengths (modules).
6. Define a scalar product of vectors as the sum of works of their abscissae, ordinates and z-coordinates: N1•N2 = k1 • k2 + l1 • l2 + n1 • n2.
7. Calculate square roots from the sums of squares of coordinates to define modules of the directing vectors: |N1| = √ (k1² + l1² + n1²); |N2| = √ (k2² + l2² + n2²).
8. Use all received expressions to write down the general formula of a cosine of the angle N1N2: cos (N1N2) = (k1 • k2 + l1 • l2 + n1 • n2) / (√ (k1² + l1² + n1²) • √ (k2² + l2² + n2²)). To find the size of the corner, count arccos from this expression.
9. Example: to define a corner between the set straight lines: L1: (x - 4)/1 = (y + 1) / (-4) = z/1; L2: x/2 = (y - 3) / (-2) = (z + 4) / (-1).
10. Decision: N1 = (1,-4, 1); N2 = (2,-2,-1).N1•N2 = 2 + 8 – 1 = 9; |N1| • |N2| = 9 · √2.cos (N1N2) = 1 / √ 2 → N1N2 = π/4.