The given line segment has endpoints
(3,2,−1) and
(6,2,−2). We can find the coordinates of point
P by using the section formula, which is applied here in its simplest form since
P is somewhere on the line segment directly between the two given points.
Let the coordinates of
P be
(x,y,z). Since the
x-coordinate of
P is given as 5 , and we know that
P lies on the line segment, we can use the formula for a point dividing a line segment in a given ratio (in this case, since the
x-coordinates are increasing from 3 to 6, and 5 lies two-thirds of the way from 3 to 6 , the division will be in the ratio
1:2 ).
The formula for a point
P dividing the line segment with endpoints
(x1,y1,z1) and
(x2,y2,z2) in the ratio
m:n is:
‌x=‌‌y=‌‌z=‌Substituting the given values, with
m=1 and
n=2 (as derived from the distances for the
x-coordinates, where 5 is 2 units from 3 and 1 unit from 6
), the endpoints are
(3,2,−1) and
(6,2,−2), respectively. Therefore, the
y-coordinate of
P would be calculated as:
y=‌=‌=‌=2Since the calculation for
y straightforwardly results in 2, and the
y-coordinates of both endpoints of the segment are 2 , the
y coordinate of
P remains constant throughout the line segment at 2. Thus, the correct answer is: