Question

Let L1 be the line through the points A(3,1,-1) and B(1,3,0), and L2 be the line through the points P(-1,2,3) and Q(2,1,5). Find the shortest distance between these two lines.

Answer 1

To find the shortest distance between two lines, we can use the formula involving vectors. First, we find a vector normal to both lines by taking the cross product of their directional vectors. Then, we can calculate the distance between the two lines using the formula: d = |AP · n| / |n|, where AP is the vector from a point on one line to the other line and n is the normal vector.

Step-by-step explanation:

To find the shortest distance between two lines, we can use the formula involving vectors. First, we find a vector normal to both lines by taking the cross product of their directional vectors. Then, we can calculate the distance between the two lines using the formula: d = |AP · n| / |n|, where AP is the vector from a point on one line to the other line and n is the normal vector.

In this case, the directional vectors for the lines are AB = (-2, 2, 1) and PQ = (3, -1, 2). Taking the cross product, we find n = (-3, -1, -5).

Next, we choose a point on one line, say A(3, 1, -1), and find the vector AP = P - A = (-4, 1, 4). Finally, we can substitute these values into the formula to find the shortest distance: d = |(-4, 1, 4) · (-3, -1, -5)| / |(-3, -1, -5)| = 5 units.