Coordinate geometry in 3-dimensions extends the concepts of geometry into a space that includes length, width, and height. This space is defined by the three-dimensional Cartesian coordinate system, composed of three axes: the x-axis (horizontal), y-axis (vertical), and z-axis (depth).
The 3D coordinate system allows us to specify the location of points in three-dimensional space using ordered triples \((x, y, z)\), where \(x\) represents the position along the x-axis, \(y\) along the y-axis, and \(z\) along the z-axis. The origin, denoted as \((0, 0, 0)\), is the point where all three axes intersect.
The distance \(d\) between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in 3D space can be calculated using the formula:
\(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\)
A line in 3D can be defined using parametric equations involving a point on the line \((x_0, y_0, z_0)\) and the direction vector \(\vec{v} = (a, b, c)\). The parametric equations are:
\(x = x_0 + at\)
\(y = y_0 + bt\)
\(z = z_0 + ct\)
where \(t\) is a parameter that varies over the real numbers.
A plane in 3D space can be defined by an equation of the form:
\(Ax + By + Cz + D = 0\)
where \(A\), \(B\), \(C\), and \(D\) are constants, and \(x\), \(y\), and \(z\) are the coordinates of any point on the plane.
The angle \(\theta\) between two lines with direction vectors \(\vec{v_1} = (a_1, b_1, c_1)\) and \(\vec{v_2} = (a_2, b_2, c_2)\) can be found using the dot product formula:
\(\cos{\theta} = \frac{\vec{v_1} \cdot \vec{v_2}}{\lVert \vec{v_1} \rVert \lVert \vec{v_2} \rVert} = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}}\)
where \(\lVert \vec{v_i} \rVert\) denotes the magnitude of vector \(\vec{v_i}\).
The shortest distance \(d\) from a point \((x_0, y_0, z_0)\) to a plane \(Ax + By + Cz + D = 0\) can be found using the formula:
\(d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}\)
A line defined by parametric equations and a plane can intersect at a point, be parallel (no intersection), or the line can lie on the plane. To find the point of intersection (if it exists), substitute the parametric equations of the line into the plane equation and solve for the parameter \(t\).
The 3D coordinate system allows for the calculation of volume and surface area of geometric solids such as spheres, cylinders, and pyramids. For example, the volume \(V\) of a sphere with radius \(r\) is:
\(V = \frac{4}{3}\pi r^3\)
and the surface area \(A\) is:
\(A = 4\pi r^2\)
Coordinate geometry in 3D has vast applications in fields such as engineering, astronomy, physics, and computer graphics. It aids in modeling real-world objects, understanding their properties, and visualizing complex systems.
Consider two points, \(P_1(1, 2, 3)\) and \(P_2(4, 5, 6)\). To find the distance between them, we apply the distance formula:
\(d = \sqrt{(4-1)^2 + (5-2)^2 + (6-3)^2} = \sqrt{3^2 + 3^2 + 3^2}\)
Given three points on a plane \(A(1, 0, 0)\), \(B(0, 1, 0)\), and \(C(0, 0, 1)\), we can determine the plane's equation by solving for \(A\), \(B\), \(C\), and \(D\). One such plane that passes through these points is \(x + y + z - 1 = 0\).
Coordinate geometry in 3-dimensions expands upon the principles of shapes, measurements, and equations from two dimensions to three. Understanding the three-dimensional coordinate system, along with equations and concepts associated with points, lines, planes, and solids, provides foundational knowledge for exploring more complex geometric and physical concepts in real-world applications.
