Answer:
Explanation:
a. To determine a vector equation for the line passing through points A(1,2,4) and B(2,3,6), we can find the direction vector of the line by subtracting the coordinates of the two points.
Direction vector:
d = B - A = (2, 3, 6) - (1, 2, 4) = (1, 1, 2)
Now, we can express the vector equation for the line as:
r = A + td
where r is a position vector on the line, t is a parameter, A is a point on the line (A(1,2,4)), and d is the direction vector we found.
The vector equation for the line is: r = (1,2,4) + t(1,1,2)
b. To determine the respective parametric equations of the line, we can assign variables to each coordinate of the point A and the direction vector.
Let x = 1 + t, y = 2 + t, and z = 4 + 2t.
The respective parametric equations of the line are:
x = 1 + t
y = 2 + t
z = 4 + 2t
c. The vector equation of the line in parametric form is r = (1,2,4) + t(1,1,2).
To write the equation in non-parametric form, we can express x, y, and z in terms of t:
x = 1 + t
y = 2 + t
z = 4 + 2t
Rearranging the equations, we can eliminate t:
t = x - 1
t = y - 2
t = (z - 4)/2
Equating the expressions for t, we have:
x - 1 = y - 2 = (z - 4)/2
This is the non-parametric equation of the line.
In summary:
a. Vector equation for the line: r = (1,2,4) + t(1,1,2)
b. Parametric equations of the line: x = 1 + t, y = 2 + t, z = 4 + 2t
c. Vector equation of the line in parametric form: r = (1,2,4) + t(1,1,2)
Non-parametric equation of the line: x - 1 = y - 2 = (z - 4)/2