Line of Intersection of Two Planes Calculator

A plane in three-dimensional space is defined by the linear equation $Ax + By + Cz = D$. When two distinct, non-parallel planes intersect, they do so along a straight line. The mathematical process relies on vector cross products and linear equation solving. First, the normal vectors of each plane, given by $\vec{n}_1 = \langle A_1, B_1, C_1 \rangle$ and $\vec{n}_2 = \langle A_2, B_2, C_2 \rangle$, are identified. The direction vector of the resulting line ($\vec{v}$) is perpendicular to both normal vectors and is found via the cross product: $\vec{v} = \vec{n}_1 \times \vec{n}_2$. This yields components $(B_1C_2 - C_1B_2, C_1A_2 - A_1C_2, A_1B_2 - B_1A_2)$. Next, a specific point on the line is located by setting one coordinate (such as $x$, $y$, or $z$) to zero and solving the resulting system of two linear equations for the remaining two variables.

Consider finding the line of intersection for Plane 1: $2x + 3y + z = 6$ and Plane 2: $x - y + 2z = 4$. Here, our coefficients are $A_1=2, B_1=3, C_1=1, D_1=6$ and $A_2=1, B_2=-1, C_2=2, D_2=4$. First, we compute the direction vector components by taking the cross product of the normal vectors: $v_x = (3)(2) - (1)(-1) = 7$, $v_y = (1)(1) - (2)(2) = -3$, and $v_z = (2)(-1) - (3)(1) = -5$. Thus, the direction vector is $\langle 7, -3, -5 \rangle$. To find a point on the line, we can set $z = 0$ and solve $2x + 3y = 6$ and $x - y = 4$. Multiplying the second equation by 2 gives $2x - 2y = 8$. Subtracting this from the first equation yields $5y = -2$, so $y = -0.4$. Substituting back gives $x = 3.6$. Therefore, the line passes through the point $(3.6, -0.4, 0)$ with direction $\langle 7, -3, -5 \rangle$.

When working with plane intersections, always check if the normal vectors are scalar multiples of one another. If they are, the planes are strictly parallel and will never intersect, resulting in an empty set. If both the normal vectors and the constant terms are proportional, the two equations describe the exact same plane, meaning they intersect along an entire infinite planar surface rather than a single line. Always simplify your direction vector by dividing by its greatest common divisor (GCD) to keep your final coordinates and ratios in their cleanest reduced form.