Final answer:
To find the distance from a point to a line, we can use the formula d = |a × b| / |a|, where a is the vector along the line and b is the vector from the point to the line. Using the given line x = 2t, y = 5 − 2t, z = 1 + t and the point (0, 1, 1), we can find the distance step-by-step.
Step-by-step explanation:
To find the distance from the point (0, 1, 1) to the line x = 2t, y = 5 − 2t, z = 1 + t, we can use the formula d = |a × b| / |a|, where a = QR and b = QP.
We first need to find the vector a. The two points Q and R on the line can be written as Q(2t, 5 − 2t, 1 + t) and R(0, 1, 1). So, a = QR = R - Q = (0, 1, 1) - (2t, 5 - 2t, 1 + t) = (-2t, -4 + 2t, -t).
Next, we find the vector b. The point P is given as (0, 1, 1), so b = QP = P - Q = (0, 1, 1) - (2t, 5 - 2t, 1 + t) = (-2t, 4 - 2t, t).
Now, we can calculate the cross product of a and b: a × b = (-2t, -4 + 2t, -t) × (-2t, 4 - 2t, t) = (2t^2, 0, -4t^2 - 4t).
Finally, we can find the distance by calculating the magnitude of the cross product and dividing it by the magnitude of a: d = |a × b| / |a| = |(2t^2, 0, -4t^2 - 4t)| / |(-2t, -4 + 2t, -t)| = √(4t^4 + (4t^2 + 4t)^2) / √(4t^2 + (-4t + 4)^2).
This is the formula for the distance from the point to the given line.