Write a vector equation AND parametric equations of the line through the points A(6,-1,5) and B(-2,-3,6)
direction vector = [8,2,-1]
a vector equation
r = (6,-1,5) + t(8,2,-1)
use that to write the parametrics
how do i write the parametrics
To find the vector equation and parametric equations of the line through points A(6, -1, 5) and B(-2, -3, 6), you need to consider the vector that represents the direction of the line and a point on the line.
Vector Equation:
Let's start by finding the direction vector of the line. The direction vector can be obtained by subtracting the coordinates of point A from the coordinates of point B:
Vector AB = B - A = (-2, -3, 6) - (6, -1, 5) = (-2 - 6, -3 - (-1), 6 - 5) = (-8, -2, 1)
Now, let's choose point A (6, -1, 5) to be a point on the line. We can write the vector equation of the line as:
r = A + t * AB,
where r represents the position vector of any point on the line, and t is the scalar parameter.
Therefore, the vector equation of the line is:
r = (6, -1, 5) + t * (-8, -2, 1)
Parametric Equations:
To write the parametric equations, we'll need to break down the vector equation into three component equations representing x, y, and z coordinates.
x = 6 - 8t
y = -1 - 2t
z = 5 + t
Thus, the parametric equations of the line are:
x = 6 - 8t
y = -1 - 2t
z = 5 + t
These equations allow you to find the coordinates of any point on the line by plugging in different values of the parameter t.