find a set of parametric equations for the line that passes through the given points.
(-9, 5, 1), (2, -3, 8)
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To find a set of parametric equations for the line passing through the given points (-9, 5, 1) and (2, -3, 8), we need to determine the direction vector of the line and an initial point on the line.
The direction vector can be found by subtracting the coordinates of the initial point from the coordinates of the terminal point:
Direction vector = (2, -3, 8) - (-9, 5, 1)
= (2 + 9, -3 - 5, 8 - 1)
= (11, -8, 7)
Now, let's choose one of the given points, say (-9, 5, 1), as the initial point on the line.
Therefore, the set of parametric equations for the line passing through the given points is:
x = -9 + 11t
y = 5 - 8t
z = 1 + 7t
where 't' is a parameter that can take any real value.
To find a set of parametric equations for the line that passes through the given points (-9, 5, 1) and (2, -3, 8), we can use the following steps:
Step 1: Identify the direction vector for the line.
To find the direction vector for the line, we need to subtract the coordinates of one point from the coordinates of the other point. Let's subtract the coordinates of (-9, 5, 1) from the coordinates of (2, -3, 8):
Direction vector = (2 - (-9), -3 - 5, 8 - 1)
= (11, -8, 7)
So, the direction vector of the line is (11, -8, 7).
Step 2: Express the parametric equations using the direction vector.
The parametric equations for a line in three dimensions can be expressed as:
x = x₀ + at
y = y₀ + bt
z = z₀ + ct
where (x, y, z) are the coordinates of any point on the line, (x₀, y₀, z₀) are the coordinates of a known point on the line, (a, b, c) are the components of the direction vector, and t is a parameter that can take any real value.
Let's choose (-9, 5, 1) as the known point on the line, and (11, -8, 7) as the direction vector.
x = -9 + 11t
y = 5 - 8t
z = 1 + 7t
Therefore, the set of parametric equations for the line passing through the given points (-9, 5, 1) and (2, -3, 8) is:
x = -9 + 11t
y = 5 - 8t
z = 1 + 7t