Calculate the requested distance.
-The distance from the point S(7, 6, -8) to the line x = -2 + 4t, y = -9 + 3t, z = 3 + 12t
A) $\frac { \sqrt { 69,562 } } { 13 }$
B) $\frac { 69562 } { 169 }$
C) $\frac { 69562 } { 13 }$
D) $\frac { \sqrt { 69,562 } } { 169 }$

Question

Calculate the requested distance.
-The distance from the point S(7, 6, -8) to the line x = -2 + 4t, y = -9 + 3t, z = 3 + 12t 
A) $\frac { \sqrt { 69,562 } } { 13 }$
B) $\frac { 69562 } { 169 }$
C) $\frac { 69562 } { 13 }$
D) $\frac { \sqrt { 69,562 } } { 169 }$

Quizplus · Accepted Answer

To find the distance from a point to a line, we can use the formula:

distance = |(P1-P0) x v| / |v|

where P1 is the point on the line closest to the given point S, P0 is the given point S, v is the direction vector of the line.

First, we need to find a point P1 on the line. We can do this by substituting the given formula for the line:

x = -2 + 4t
y = -9 + 3t
z = 3 + 12t

into another formula for finding the point on a line closest to a given point:

P1 = P0 + [(P0-Q) dot v / |v|^2] v

where Q is any point on the line, and dot represents the dot product.

Let's use Q = (-2, -9, 3) (the point on the line when t = 0):

P1 = (7, 6, -8) + [(7-(-2), 6-(-9), -8-3) dot (4, 3, 12) / (4^2 + 3^2 + 12^2)] (4, 3, 12)

P1 = (7, 6, -8) + (9, 15, -11) / 169 * (4, 3, 12)

P1 = (19/13, 117/169, 115/169)

Now we can plug this into the distance formula:

distance = |(P1-P0) x v| / |v|

distance = |(19/13 - 7, 117/169 - 6, 115/169 + 8) x (4, 3, 12)| / sqrt(4^2+3^2+12^2)

distance = (12/13, 315/169, 303/169) / sqrt(169)

distance = sqrt(144 + 315^2/169 + 303^2/169) / 13

distance = sqrt(69562) / 13

Therefore, the answer is A: $\frac { \sqrt { 69,562 } } { 13 }$.

Calculate the Requested Distance 69,56213\frac { \sqrt { 69,562 } } { 13 }1369,562​​

Calculate the Requested Distance $\frac { \sqrt { 69,562 } } { 13 }$