Section Formula In Three Dimensions

Question

A point divides the segment joining (2, -1, 3) and (4, 3, -1) internally in the ratio 1:3, so find that point.

RankGuru · Accepted Answer

The technique is the three-dimensional section formula: a point dividing the join of (x_1, y_1, z_1) and (x_2, y_2, z_2) internally in ratio m:n has coordinates ((m x_2 + n x_1)/(m+n), (m y_2 + n y_1)/(m+n), (m z_2 + n z_1)/(m+n)). This weighted-average formula is a core JEE Advanced result. With m:n = 1:3, m + n = 4, and endpoints (2, -1, 3) and (4, 3, -1): x = (1\cdot4 + 3\cdot2)/4 = (4 + 6)/4 = 10/4 = 5/2; y = (1\cdot3 + 3\cdot(-1))/4 = (3 - 3)/4 = 0; z = (1\cdot(-1) + 3\cdot3)/4 = (-1 + 9)/4 = 8/4 = 2. So the point is (5/2, 0, 2). Option (3, 1, 1) is the midpoint, ratio 1:1. Option (7/2, 2, 0) uses ratio 3:1, swapping m and n. Option (2/3, -1/3, 1) divides by the wrong total. This applies the internal section-formula theorem. Plausibility check: the point lies between the two endpoints since each coordinate falls within the corresponding endpoint range, consistent with internal division.

Section Formula In Three Dimensions

Select the correct option:

🔒 Solution Hidden from View

More three dimensional geometry Practice Questions

Decide whether the three points (1, 2, 3), (2, 4, 6), and (4, 8, 12) are collinear in three-dimensio...

Find the radius of the sphere whose equation is x^2 + y^2 + z^2 - 2x + 4y - 6z + 5 = 0 in three-dime...

Find the acute angle between the two planes represented by the equations x + y + z = 5 and x - y + 2...

Examine whether the lines (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-2)/3 = (y-3)/4 = (z-4)/5 are coplanar.

A family of planes passes through the line of intersection of x + y + z = 1 and 2x + 3y - z = 4 in s...

Two skew lines pass through (1, 2, 3) and (2, 4, 5) with direction vectors (1, 0, -1) and (0, 1, 1) ...

Determine the perpendicular distance from the point (1, 2, -1) to the plane whose equation is given ...

Given the point (1, 0, 0) and its reflection across the plane x + y + z = 3, identify the image poin...

Suppose a plane passes through the three non-collinear points (1, 0, 0), (0, 1, 0), and (0, 0, 1) in...

A directed line segment joins the point (3, 4, 5) to the point (5, 6, 7), so what are its direction ...

Locate the foot of the perpendicular drawn from the point (1, 2, 3) onto the line given by x/1 = y/2...

Evaluate the perpendicular distance between the two parallel planes 2x - y + 2z = 5 and 2x - y + 2z ...

About This Question

Solution