To determine the angle between two straight in space even if they do not intersect. In this case, you need to mentally combine the start of their directing vectors and calculate the magnitude of the resulting angle. In other words, is any of the adjacent angles formed by intersecting straightdrawn parallel data.
There are several ways to specify a straight line in space, for example, vector-parametric, parametric and canonical. The three mentioned method is convenient to use when finding the angle, because all of them require the introduction of coordinates of directing vectors. Knowing these values, you can define an educated angle cosine theorem of vector algebra.
Suppose two lines L1 and L2 is set to the canonical equations:L1: (x – x1)/k1 = (y – y1)/l1 = (z – z1)/n1;L2: (x – x2)/k2 = (y – y2)/l2 = (z – z2)/n2.
Using the values of ki, li and ni, write down the coordinates of directing vectors of the straight. Call them N1 and N2:N1 = (k1, l1, n1);N2 = (k2, l2, n2).
The formula for the cosine of the angle between the vectors represents the ratio between their dot product and the result of arithmetic multiplication of their lengths (modules).
Define the scalar product as the sum of the products of their x, y and applicator that con:N1•N2 = k1•k2 + l1•l2 + n1•n2.
Calculate the square roots of the sums of the squares of the coordinates to define the modules of the guides of the vectors:|N1| = √(k12 + l12 + n12);|N2| = √(k22 + l22 + n22).
Use all the expressions to write the General formula of the cosine of the angle N1N2:cos (N1N2) = (k1•k2 + l1•l2 + n1•n2)/( √(k12 + l12 + n12)•√(k22 + l22 + n22)).To find the value of the angle, calculate the arccos of this expression.
Example: determine the angle between the given straight:L1: (x - 4)/1 = (y + 1)/(-4) = z/1;L2: x/2 = (y - 3)/(-2) = (z + 4)/(-1).
Solution: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.