Question

A (1, 8, 4), B (0, -11, 4), C (2,-3, 1) are three points and L is the foot of the perpendicular from A to BC. Find the coordinates of L.

Answer 1

We will be using the Cartesian equation of line passing through two points using the formula$\dfrac{{x - {x_1}}}{{{x_2} - {x_1}}} = \dfrac{{y - {y_1}}}{{{y_2} - {y_1}}} = \dfrac{{z - {z_1}}}{{{z_2} - {z_1}}}$. As L is the foot of the perpendicular from point A to BC, we need to find the direction ratio of AL. On solving we get the value of the variable. On substituting the value of variable gives the coordinates of the foot of the perpendicular.

Complete step-by-step answer:
Given:
The three given points are A (1, 8, 4), B (0, -11, 4), C (2,-3, 1)
L is the foot of the perpendicular from A to BC
The Cartesian equation of a line passing through B (0, -11, 4) and C (2,-3, 1) is,
$\dfrac{{x - {x_1}}}{{{x_2} - {x_1}}} = \dfrac{{y - {y_1}}}{{{y_2} - {y_1}}} = \dfrac{{z - {z_1}}}{{{z_2} - {z_1}}}$
Therefore, equation of line BC is,
$\Rightarrow \dfrac{{x - 0}}{{2 - 0}} = \dfrac{{y + 11}}{{ - 3 + 11}} = \dfrac{{z - 4}}{{1 - 4}}$
$\Rightarrow \dfrac{x}{2} = \dfrac{{y + 11}}{8} = \dfrac{{z - 4}}{{ - 3}}$
Consider the diagram shown below. Let L (x, y, z) be the foot of the perpendicular from point A (1, 8, 4) to the given line.
The coordinates of point L on the line BC is given by
$\Rightarrow \dfrac{x}{2} = \dfrac{{y + 11}}{8} = \dfrac{{z - 4}}{{ - 3}} = \lambda$
$\Rightarrow x = 2\lambda$
$\Rightarrow y + 11 = 8\lambda$ $\Rightarrow y = 8\lambda - 11$
$\Rightarrow z = - 3\lambda + 4$
The direction ratio of AL is (${x_2} - {x_1}$), (${y_2} - {y_1}$), (${z_2} - {z_1}$).
$\Rightarrow \left( {2\lambda - 1} \right),\left( {8\lambda - 11 - 8} \right),\left( { - 3\lambda + 4 - 4} \right)$
$\Rightarrow \left( {2\lambda - 1} \right),\left( {8\lambda - 19} \right),\left( { - 3\lambda } \right)$
Since, both the line are perpendicular, we know that
${a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0$
$\Rightarrow 2\left( {2\lambda - 1} \right) + 8\left( {8\lambda - 19} \right) + \left( { - 3} \right)\left( { - 3\lambda } \right) = 0$
$\Rightarrow 4\lambda - 2 + 16\lambda - 152 + 9\lambda = 0$
$\Rightarrow 77\lambda = 154$
$\Rightarrow \lambda = \dfrac{{154}}{{77}}$
$\Rightarrow \lambda = 2$
Therefore, coordinates of L are,
$x$= 2(2) = 4
$y$= 8(2) – 11 = 5
$z$= -3(2) + 4 = -2
Hence, the coordinates of the foot of the perpendicular is (4, 5, -2).

Note: The perpendicular foot, also called the foot of an altitude, is the point on the leg opposite a given vertex of a triangle at which the perpendicular passing through that vertex intersects the side. The length of the line segment from the vertex to the perpendicular foot is called the altitude of the triangle. When a line is drawn from a point to a plane, its intersection with the plane is known as the foot.