Is the line through (-4, -6, 1) and (-2, 0, -3) parallel to the line through (10, 18, 4) and (5, 3, 14)?

Three-dimensional Geometry is one of the most important topics in mathematics. It deals with curves and planes in the three-dimensional plane. Vectors and direction cosines are extensively used to represent the lines and planes. Let's find out how to check if two lines are parallel to each other, in this blog.

Answer: The line through (-4, -6, 1) and (-2, 0, -3), is parallel to the line through (10, 18, 4) and (5, 3, 14).

Let us understand how we arrived at the solution.

Explanation:

Step 1: First, we need to calculate the direction cosines of the lines.

⇒The First line has direction cosines: (-2, -6, 4)

⇒The Second line has direction cosines: (5, 15, -10)

Step 2: Next, we check if the lines are parallel. For that, we need to check if the direction cosines of one line are multiples of the corresponding direction cosines of the other line.

For x-direction, the ratio is -2 / 5.

For y-direction, the ratio is -6 / 15 = -2 / 5.

For z-direction, the ratio is 4 / -10 = -2 / 5.

Hence, direction cosines of the first line = 2 / 5 times direction cosines of the Second line.

⇒ From the above statements, we see that the direction cosines are in the ratio of -2 / 5.

⇒ Since the ratio is the same, we can say that the  given lines are parallel to each other

Hence, we come to the conclusion that the line through (-4, -6, 1) and (-2, 0, -3), is parallel to the line through (10, 18, 4) and (5, 3, 14).