Two lines in intersecting planes are ____skew?
always sometimes or never
Two parallel Lines are ________ Coplanar
Two lines in intersecting planes can be either parallel or skew.
To determine if the lines are skew, we need to check if they are non-intersecting and not parallel.
1. First, find the direction vectors of both lines. This can be done by subtracting the coordinates of two points on each line. Let's call these direction vectors vector A and vector B.
2. Next, check if the direction vectors are parallel by calculating the scalar triple product of vector A, vector B, and the normal vector to one of the planes. If the scalar triple product equals zero, the lines are parallel.
Example:
Let's say we have two lines:
Line 1: r = (1, 2, 3) + t(2, -1, 4)
Line 2: r = (2, 0, 1) + s(1, -1, 2)
1. Calculate the direction vectors:
Direction vector of Line 1 = (2, -1, 4)
Direction vector of Line 2 = (1, -1, 2)
2. Calculate the scalar triple product:
Let's choose the normal vector of Plane 1 = (0, 1, -1)
Scalar triple product = (2, -1, 4) . (1, -1, 2) x (0, 1, -1)
where "." represents the dot product and "x" represents the cross product.
If the scalar triple product is zero, the lines are parallel. If it is non-zero, the lines are skew.
Note: Skew lines are non-intersecting lines that are not parallel.