Final answer:
To find the parametric and symmetric equations of a line passing through two points, subtract the corresponding coordinates to find the direction ratios. Then, use the formula x = x1 + at, y = y1 + bt, and z = z1 + ct to write the parametric equations. The symmetric equations can be written as (x - x1)/a = (y - y1)/b = (z - z1)/c.
Step-by-step explanation:
To find the parametric and symmetric equations of a line passing through two points, we can use the formula:
x = x1 + at
y = y1 + bt
z = z1 + ct
Where (x1, y1, z1) are the coordinates of one point, (x, y, z) are the coordinates of any point on the line, (a, b, c) are the direction ratios or components of the line, and t is a parameter. Let's work through an example:
Given points A(0, 0.5, 1) and B(2, 1, -3), we can find the direction ratios by subtracting the corresponding coordinates: a = 2 - 0 = 2, b = 1 - 0.5 = 0.5, c = -3 - 1 = -4. The parametric equations are:
x = 0 + 2t
y = 0.5 + 0.5t
z = 1 - 4t
In symmetric form, the equations can be written as:
(x - 0)/2 = (y - 0.5)/0.5 = (z - 1)/-4