Final answer:
To find the distance of a point from a line, we need to find the perpendicular distance from the point to the line. In this case, the line is represented by the parametric equation r(t) = (-1, 3t, -1, 4t, 3-3t) and the point is (4, 3, -4). We can use the formula for the distance between a point and a line to solve this problem.
Step-by-step explanation:
To find the distance of a point from a line, we need to find the perpendicular distance from the point to the line. In this case, the line is represented by the parametric equation r(t) = (-1, 3t, -1, 4t, 3-3t) and the point is (4, 3, -4). We can use the formula for the distance between a point and a line to solve this problem.
Let's start by finding a point on the line and the direction vector of the line. Taking t = 0, we get the point (-1, 0, -1, 0, 3). Taking t = 1, we get the point (-1, 3, -1, 4, 0). Subtracting these two points gives us the direction vector of the line: (0, 3, 0, 4, -3).
Next, we need to find a vector from any point on the line to the given point (4, 3, -4). Subtracting the coordinates of these two points gives us the vector (4-(-1), 3-0, -4-(-1), 3-0, -4-(3)) = (5, 3, -3, 3, -7).
The formula for the distance between a point and a line is given by: d = |(P - Q) - ((P - Q)·v) * v| / |v|, where P is a point on the line, Q is the given point, v is the direction vector of the line, · denotes the dot product, and |·| denotes the magnitude of a vector.
Substituting the values into the formula, we have: d = |(5, 3, -3, 3, -7) - ((5, 3, -3, 3, -7)·(0, 3, 0, 4, -3)) * (0, 3, 0, 4, -3)| / |(0, 3, 0, 4, -3)|.
Calculating the magnitudes and dot product, we have: d = |(5, 3, -3, 3, -7) - (-37) * (0, 3, 0, 4, -3)| / |(0, 3, 0, 4, -3)| = |(5, 3, -3, 3, -7) + (0, -37, 0, -148, 111)| / |(0, 3, 0, 4, -3)| = |(5, -34, -3, -145, 104)| / |(0, 3, 0, 4, -3)|.
Finally, calculating the magnitudes, we have: d = sqrt(5^2 + (-34)^2 + (-3)^2 + (-145)^2 + 104^2) / sqrt(0^2 + 3^2 + 0^2 + 4^2 + (-3)^2).
Therefore, the distance of the point (4, 3, -4) from the line r(t) = (-1, 3t, -1, 4t, 3-3t) is equal to sqrt(18690) / sqrt(34). We can further simplify this by rationalizing the denominator.
Thus, the distance is approximately 26.843 units.