Calculate the requested distance.
-The distance from the point S(-6, 1, -4) to the line x = -2 + 2t, y = -1 + 6t, z = 6 + 9t
A) $\frac { 2 \sqrt { 1781 } } { 11 }$
B) $\frac { 7124 } { 121 }$
C) $\frac { 7124 } { 11 }$
D) $\frac { 2 \sqrt { 1781 } } { 121 }$

Question

Calculate the requested distance.
-The distance from the point S(-6, 1, -4) to the line x = -2 + 2t, y = -1 + 6t, z = 6 + 9t 
A) $\frac { 2 \sqrt { 1781 } } { 11 }$
B) $\frac { 7124 } { 121 }$
C) $\frac { 7124 } { 11 }$
D) $\frac { 2 \sqrt { 1781 } } { 121 }$

Quizplus · Accepted Answer

1. First, find a vector parallel to the line: - The direction vector is <2, 6, 9> 2. Find the vector connecting a point on the line to point S: - Let's use t=0 for simplicity, then the vector is <(-2-(-6)), (-1-1), (6-(-4))> = <4, -2, 10> 3. Use the projection formula to find the distance: - Distance = |proj_u(v)|, where u is the unit vector in the direction of the line and v is the vector found in step 2. - u = <2/(\sqrt{185}), 6/(\sqrt{185}), 9/(\sqrt{185})> (normalize the direction vector) - proj_u(v) = <4, -2, 10> · <2/(\sqrt{185}), 6/(\sqrt{185}), 9/(\sqrt{185})> * <2, 6, 9>/(\sqrt{185}) = (8/(\sqrt{185}) - 12/(\sqrt{185}) + 90/(\sqrt{185}))/(\sqrt{185}) = 86/(\sqrt{185}) - Distance = |proj_u(v)| = 86/(\sqrt{185}) = (2\sqrt{1781})/11 (after rationalizing the denominator)

Calculate the Requested Distance 2178111\frac { 2 \sqrt { 1781 } } { 11 }1121781​​

Calculate the Requested Distance $\frac { 2 \sqrt { 1781 } } { 11 }$