Equation Of A Plane Through Three Points

Question

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

RankGuru · Accepted Answer

The relevant idea is the intercept form of a plane, x/a + y/b + z/c = 1, which directly applies when a plane cuts the axes at known intercepts; equivalently one can use the determinant form through three points. This is a classic JEE Advanced shortcut. The given points are precisely the unit intercepts on the x, y and z axes, so a = b = c = 1, yielding x/1 + y/1 + z/1 = 1, that is x + y + z = 1. To verify by the normal approach, vectors in the plane are (0,1,0)-(1,0,0) = (-1, 1, 0) and (0,0,1)-(1,0,0) = (-1, 0, 1); their cross product is (1, 1, 1), a valid normal, and using point (1,0,0) gives 1(x-1)+1y+1z = 0, i.e. x + y + z = 1. Option x + y + z = 0 fails the points. Option x + y + z = 3 misplaces the constant. Option x - y + z = 1 uses a wrong normal. This rests on the intercept-form plane theorem. Plausibility check: each of the three given points individually satisfies x + y + z = 1, confirming the plane indeed contains them all.

Equation Of A Plane Through Three Points

Select the correct option:

🔒 Solution Hidden from View

More three dimensional geometry Practice Questions

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

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...

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