Write the symmetric equations of the line passing through A(3, 2, -1) and B(2, -1, -2)
the direction vector is <1, 3, 1>
Using the point (3,2,-1) we have ....
(x-3)/1 = (y-2)/3 = (z+1)/1
To find the symmetric equations of a line passing through two given points, you need to find the direction vector of the line and an equation of the line in terms of the parameter t.
Step 1: Find the direction vector
To find the direction vector, subtract the coordinates of one point from the coordinates of the other point.
Direction vector = AB = B - A = (2 - 3, -1 - 2, -2 - (-1)) = (-1, -3, -1)
Step 2: Write the symmetric equations
The symmetric equations of a line are written as:
x = x₀ + at
y = y₀ + bt
z = z₀ + ct
where (x₀, y₀, z₀) is a point on the line, and (a, b, c) is the direction vector.
Choose one of the points, A(3, 2, -1), to obtain the specific symmetric equations. We can write:
x = 3 + (-1)t
y = 2 + (-3)t
z = -1 + (-1)t
Thus, the symmetric equations of the line passing through points A(3, 2, -1) and B(2, -1, -2) are:
x = 3 - t
y = 2 - 3t
z = -1 - t