1. a) Write the vector and parametric equations of the line through the points A(6, -1, 5) and B(-2, -3, 6).
b) Find another point on the line in (a).
r = 6i-j+5k + t(-8i-2j+k)
x = 6-8t
y = -1-2t
z = 5+t
Can you please do b) Steve? Thank you.
Steve, for a) I got this instead:
vector equation:
[x, y, z] = [6, -1, 5] + t[-8, -2, 1]
parametric equation:
x=6+t(-2)
x=6-2t
y=-1+t(-3)
y=-1-3t
z=5+6t
is that correct? ...
a) To find the vector equation of the line passing through points A(6, -1, 5) and B(-2, -3, 6), we need to find the direction vector of the line and a point on the line.
The direction vector of the line can be found by taking the difference between the coordinates of the two points:
Direction vector = B - A = (-2, -3, 6) - (6, -1, 5) = (-8, -2, 1)
Now we can write the vector equation of the line as:
R = A + t * Direction vector
where R is a position vector on the line and t is a scalar parameter.
Substituting the values, the vector equation becomes:
R = (6, -1, 5) + t * (-8, -2, 1)
b) To find another point on the line, we can choose any value for the parameter t and substitute it into the vector equation. Let's choose t = 1:
R = (6, -1, 5) + 1 * (-8, -2, 1)
= (6, -1, 5) + (-8, -2, 1)
= (6 - 8, -1 - 2, 5 + 1)
= (-2, -3, 6)
Therefore, another point on the line is (-2, -3, 6).