Calculate the requested distance.
-The distance from the point S(3, 2, -6) to the line x = -6 + 2t, y = 9 + 2t, z = -8 + 2t
A) $\frac { 1544 } { 9 }$
B) $\frac { 1544 } { 3 }$
C) $\frac { 2 \sqrt { 386 } } { 3 }$
D) $\frac { 2 \sqrt { 386 } } { 9 }$

Question

Calculate the requested distance.
-The distance from the point S(3, 2, -6) to the line x = -6 + 2t, y = 9 + 2t, z = -8 + 2t 
A) $\frac { 1544 } { 9 }$
B) $\frac { 1544 } { 3 }$
C) $\frac { 2 \sqrt { 386 } } { 3 }$
D) $\frac { 2 \sqrt { 386 } } { 9 }$

Quizplus · Accepted Answer

First, we need to find the point on the line that is closest to point S. We can use the formula for the projection of a vector onto another vector to find this point: Let's define vector a as the vector from S to any point on the line (x,y,z), i.e. a = . Let's define vector b as the direction vector of the line, i.e. b = <2, 2, 2>. The projection of vector a onto vector b is given by: proj_b a = (a dot b / ||b||^2) * b where "dot" represents the dot product, and "||b||" represents the magnitude of vector b. Plugging in the values, we get: proj_b a = ((2x-3) + (2y-2) + (2z+12)) / 12 * <2,2,2> = (2x + 2y + 2z + 1) / 12 <2,2,2> = <(x+y+z+1)/6, (x+y+z+1)/6, (x+y+z+1)/6> Now we need to find the distance between point S and this projected point. We can use the distance formula: d = ||S - proj_b a|| = ||<3,2,-6> - <(x+y+z+1)/6, (x+y+z+1)/6, (x+y+z+1)/6>|| = sqrt((3 - (x+y+z+1)/6)^2 + (2 - (x+y+z+1)/6)^2 + (-6 - (x+y+z+1)/6)^2) To minimize this distance, we can take the partial derivatives of d with respect to x, y, and z, set them equal to 0, and solve for x, y, and z. This will give us the coordinates of the point on the line that is closest to point S. Taking the partial derivative with respect to x, we get: d_x = (x+y+z+1)/6 - 3 = 0 x+y+z = 17 Taking the partial derivative with respect to y, we get: d_y = (x+y+z+1)/6 - 2 = 0 x+y+z = 11 Taking the partial derivative with respect to z, we get: d_z = (x+y+z+1)/6 + 6 = 0 x+y+z = -37 Solving this system of equations, we get x = -17, y = 9, z = -29. Now we just need to find the distance between S and this point: d = ||S - < -17, 9, -29>|| = sqrt((3 - (-17))^2 + (2 - 9)^2 + (-6 - (-29))^2) = sqrt(1544) Therefore, the answer is C.

Calculate the Requested Distance 15449\frac { 1544 } { 9 }91544​

Calculate the Requested Distance $\frac { 1544 } { 9 }$