Equations of lines in 3D

• Understanding equations of lines in 3D requires a base knowledge of vectors and their uses in describing objects in three dimensions.

• There are two integral forms for equations of lines in 3D: Parametric and Vector forms.

• Parametric Form: This representation characterizes each coordinate of points on the line in terms of a unique parameter. The equations are of the form x = x0 + at, y = y0 + bt, and z = z0 + ct.

• Vector Form: In Vector form, the equation of a line in 3D is r = a + tb, where ‘r’ represents any vector points on the line, ‘a’ is a fixed point on the line (generally the initial point), ‘b’ is a direction vector pointing in the direction of the line and ‘t’ is a scalar, giving us a sense of the magnitude and direction of the line.

• You can convert lines from Vector form to Parametric form by comparing the coefficients.

• The direction ratios of a line provide another method to write the equations of a line. These ratios, often represented by ‘a’, ‘b’, and ‘c’, are similar to the direction vector in vector form, indicating the direction of the line.

• Remember that parallel lines in 3D space have the same direction vector or ratios.

• The point of intersection for two lines in 3D space can be found by comparing the direction proportions and checking for consistency.

• It’s also important to learn about skew lines, which are lines in 3D space that do not intersect but aren’t parallel. Using vector algebra, we can also determine whether two lines in 3D are skew lines.

• It’s also advantageous to understand the concepts of perpendicular lines and the angle between two lines in 3D space.

• Practice is key to mastery in equations of lines in 3D. Apply these concepts and principles in a variety of problems for optimal understanding and skills development.