Question

A line with positive direction cosines passes through the point \(P(2,-1,2)\) and makes equal angles with the coordinate axes. The line meets the plane $$ 2 x+y+z=9 $$ at point \(Q\). The length of the line segment \(P Q\) equals (A) 1 (B) \(\sqrt{2}\) (C) \(\sqrt{3}\) (D) 2

Short answer

C) \(\sqrt{3}\)

Step-by-step solution

Understand Line Direction Cosines and Angles

A line with positive direction cosines making equal angles with the coordinate axes means that the angles between the line and each axis (x, y, z) are equal. This implies that the direction cosines of the line are equal, so if the direction cosines are denoted as l, m, and n, we have l = m = n, and since they are positive, we have l, m, n > 0. Also, since the sum of the squares of the direction cosines is equal to 1, we have l^2 + m^2 + n^2 = 1.

Find Direction Cosines

With l = m = n and using the equation l^2 + m^2 + n^2 = 1, we find the value of each direction cosine. Since there are three equal direction cosines, we have 3l^2 = 1, which means l = m = n = 1/√3. This represents the direction ratios of the line, which can be used to form the direction vector of the line (1/√3, 1/√3, 1/√3).

Parametric Equations of the Line

With the direction vector found, we can form the parametric equations of the line passing through P(2,-1,2). If t is the parameter, the equations are x = 2 + (1/√3)t, y = -1 + (1/√3)t, and z = 2 + (1/√3)t.

Find Intersection with Plane

Substitute the parametric equations into the plane equation 2x + y + z = 9 to find the value of t at the intersection point Q. 2(2 + (1/√3)t) + (-1 + (1/√3)t) + (2 + (1/√3)t) = 9 simplifies to (6/√3)t = 6, giving us t = √3.

Calculate Coordinates of Point Q

Substitute t = √3 into the parametric equations to find the coordinates of Q. Q has coordinates Q(2 + (1/√3)√3, -1 + (1/√3)√3, 2 + (1/√3)√3), which simplifies to Q(3, 0, 3).

Determine Length of PQ

Using the distance formula, we calculate the length PQ = √[(3-2)^2 + (0-(-1))^2 + (3-2)^2] = √[1 + 1 + 1] = √3.

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to use algebra to calculate geometric problems. Fundamental to coordinate geometry is the idea of representing geometric objects, such as points, lines, and planes, within a coordinate system, usually the Cartesian coordinate system.

The Cartesian system in two dimensions consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point in the plane is specified by an ordered pair of numbers known as coordinates (x, y). In three-dimensional space, this system extends by adding a z-axis, which is perpendicular to both the x and y axes, allowing us to represent points in space with a triplet of coordinates (x, y, z).

Application in Problems

Using coordinate geometry, we can address complex spatial problems by breaking them down into algebraic equations. For instance, we can describe a line in space using a point through which the line passes and its direction cosines or direction ratios, which indicate the line's orientation relative to the axes.

In the given exercise, the point P(2, -1, 2) and equal direction cosines help to establish a line's equation, which we can then analyze algebraically to determine intersections with other geometric figures like planes or other lines.

Line in Three Dimensions

A line in three-dimensional space can be perceived as an infinite set of points that extend in both directions from any point on the line. Such a line can be uniquely determined by a point through which it passes and a direction vector that indicates its orientation in space.

The direction vector is often expressed in terms of direction cosines, which are the cosines of the angles that the line makes with the positive directions of the x, y, and z axes. These angles are crucial as they illustrate how the line is oriented relative to each axis. Since the cosine function ranges from -1 to 1, the direction cosines also fall within this range.

Connecting Points and Lines

In our exercise, the line’s direction cosines are all positive and equal, suggesting that the line makes the same angle with all three coordinate axes, forming a direction vector that is equidistant from each axis. This comes in handy when we wish to describe the line's path algebraically using parametric equations. Parametric equations express the coordinates of the points on the line as expressions of a single parameter, typically denoted t, allowing us to trace the line through space.

Plane Equation Intersection

The intersection of a line and a plane in three-dimensional space is a crucial concept in coordinate geometry, allowing us to find the specific point where they cross. A plane can be represented by an equation of the form Ax + By + Cz = D, where A, B, and C are the coefficients that dictate the orientation of the plane, and D is a constant.

To determine the intersection of a line with a plane, we substitute the parametric equations of the line into the plane's equation. This process almost always yields a system of equations which can be solved to find the values of the variable that corresponds to the position where the two figures meet.

Implementation in Our Exercise

In the exercise provided, once we have the parametric equations of the line, we plug them into the plane's equation to find the value of the parameter t corresponding to point Q, the intersection. As a result, we obtain the coordinates of Q, which facilitates the calculation of the length of the line segment PQ using the distance formula, a staple tool in coordinate geometry. This exercise exemplifies how combining the concepts of three-dimensional lines and planes can solve spatial geometric problems.