Problem 13 - Precalculus with Calculus Previews

Question

Consider the point \(P(-2,5,4)\) (a) If lines are drawn from \(P\) perpendicular to the coordinate planes, what are the coordinates of the point at the base of each perpendicular? (b) If a line is drawn from \(P\) to the plane \(z=-2\), what are the coordinates of the point at the base of the perpendicular? (c) Find the point in the plane \(x=3\) that is closest to \(P\).

Step-by-Step Solution

Answer

(a) Points are (-2, 5, 0), (0, 5, 4), (-2, 0, 4); (b) Point is (-2, 5, -2); (c) Point is (3, 5, 4).

Step 1: Identify the base on the XY-plane

To find the point on the XY-plane, set the z-coordinate of point \(P\) to zero. So the point on the XY-plane is \((-2, 5, 0)\).

Step 2: Identify the base on the YZ-plane

To find the point on the YZ-plane, set the x-coordinate of point \(P\) to zero. So the point on the YZ-plane is \((0, 5, 4)\).

Step 3: Identify the base on the XZ-plane

To find the point on the XZ-plane, set the y-coordinate of point \(P\) to zero. So the point on the XZ-plane is \((-2, 0, 4)\).

Step 4: Find base on the plane z=-2

Replace the z-coordinate of point \(P\) with \(-2\) to project it perpendicularly to the plane \(z = -2\). The point is \((-2, 5, -2)\).

Step 5: Find the closest point on the plane x=3

The closest point on the plane \(x = 3\) to point \(P(-2, 5, 4)\) is found by setting the x-coordinate to 3, and keeping the y and z-coordinates the same. The closest point is \((3, 5, 4)\).

Key Concepts

Coordinate Geometry

Coordinate Geometry is an essential area of mathematics where we study geometrical shapes using a coordinate system. It helps us understand the positions and relations between points in space by using numerical coordinates. In a three-dimensional space, like the one in our original problem, each point is defined by a triple \(x, y, z\). This allows us to manipulate and study points, lines, and planes with precision.

In the problem, we have a point called \(P(-2,5,4)\). Its coordinates tell us its specific location in space: \(-2\) is the x-coordinate, \(5\) is the y-coordinate, and \(4\) is the z-coordinate. By changing these values, we can understand how the point moves relative to the coordinate axes.

Perpendicular Lines

Perpendicular lines are essential in Geometry and have a unique property: they intersect at a right angle, which is 90 degrees. In coordinate geometry, such lines can be identified by specific conditions. If two lines are perpendicular, the product of their slopes in a plane is \(-1\).

For our 3D point \(P(-2,5,4)\), we examine where perpendicular lines from this point meet each coordinate plane. This helps us identify specific points on these planes that are at right angles to the given point in space. For example, by setting one coordinate of \(P\) to zero, we project it onto the respective coordinate plane at a right angle.

Coordinate Planes

Coordinate planes are fundamental in understanding spatial geometry. These are flat surfaces defined by two axes in a coordinate system. In three-dimensional space, we deal with three major planes:

• **XY-plane** - where z is constant and usually taken as zero.
• **YZ-plane** - where x is constant, often zero.
• **XZ-plane** - where y is constant and set to zero in our exercises.

Points on these planes allow us to study their behavior and relationship with other geometrical elements. For instance, getting the point \((0, 5, 4)\) on the YZ-plane from \(P(-2,5,4)\) is done by making the x-coordinate zero.

Projection

Projection refers to the method of mapping points onto another surface or plane. It is a critical concept in geometry, allowing us to transfer information from a 3D object onto 2D surfaces like coordinate planes.

In our example, one task involved projecting point \(P(-2,5,4)\) onto the plane \(z = -2\). This is achieved by replacing the z-value of \(P\) with \(-2\), resulting in a new projected point of \((-2, 5, -2)\). Such projections help us analyze and solve geometric problems by simplifying the dimensional complexity.