Exam Questions - Parallel intersecting and skew lines

Understanding Lines

A straight line is a path of continuous, infinite points with no curves.
Parallel lines are a pair of identical lines which never intersect (cross each other) as they always maintain a constant distance from each other. In 3D geometry, the direction ratios of two parallel lines are proportional.
Intersecting lines are lines that cross at one particular point. This point is common to both lines.
Skew lines are two lines in three-dimensional space that do not intersect and are not parallel.

An important way to identify parallel lines in a plane is to look at their gradients. If two lines have equal gradients, they’re parallel. This property extends to three-dimensional space.
For intersecting lines, their slopes in two-dimension will be different. If two lines are represented in equations, these equations can be solved simultaneously to find the point of intersection.
In 3D space, skew lines will have different direction ratios and no point of intersection.

Familiarise yourself with the equations of lines. In a two-dimensional plane, lines are generally denoted by y = mx + c, where m is the slope and c is the y-intercept. In three dimensions, lines can be represented in parametric, symmetric or vector form.
When revising, pay special attention to the form of the equation that gives a line in a straight line (y = mx + c), because this will allow you to identify lines that are parallel or intersecting.

It’s crucial to be able to interpret and manipulate these equations of lines in order to identify whether they are parallel, intersecting or skew.
If you have two line equations, you can compare their components to check their relationship. If the slopes (or direction ratios in 3D) are equal, they are parallel. If the lines share a point, they intersect.

Practice on a variety of problems dealing with parallel, intersecting and skew lines. The problems could involve identifying the type of line, finding the point of intersection, or determining the slope of a line.
To identify skew lines, always attempt to solve the set of equations. If there is no solution, then the lines are skew.
For intersecting lines, remember that the point of intersection is the solution of the simultaneous equations of two lines.

In case of a problem requiring you to find the angle between two intersecting lines, remember to use the specific formula: tan θ =

(m2 - m1) / (1 + m1*m2)

, where m1 and m2 are slopes of the two given lines.

Although specific examination techniques won’t be provided, it’s important to remember that understanding the concepts and properties of parallel, intersecting and skew lines, as well as regular problem practicing, is the key to scoring well.
Lastly, don’t forget that clear presentation of your working is crucial to secure all possible method marks. Ensure your mathematical workings are clear, logical and easy to follow. Boldly demarcate your final answers from your working.