Lines and planes in 3D space are fundamental concepts in Calculus III. They help us understand how objects exist and move in three dimensions, building on our knowledge of 2D geometry.
We'll learn different ways to describe lines and planes mathematically. This includes , parametric, and symmetric equations for lines, as well as vector and scalar equations for planes. We'll also explore relationships between these objects and calculate distances.
Lines and Planes in 3D Space
Equations of lines in space
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of a line represents the line using a r, a point on the line r0, a v parallel to the line, and a scalar parameter t that varies to generate all points on the line
Example: r=⟨1,2,3⟩+t⟨4,5,6⟩
Parametric equations of a line describe the line using separate equations for x, y, and z coordinates, each involving a point on the line (x0,y0,z0), components of a direction vector (a,b,c), and the parameter t
Example: x=1+4t, y=2+5t, z=3+6t
Symmetric equations of a line express the line using ratios of differences between coordinates of a general point (x,y,z) and a specific point on the line (x0,y0,z0), divided by corresponding components of a direction vector (a,b,c)
Example: 4x−1=5y−2=6z−3
Relationships between lines
have the same direction vector or direction vectors that are scalar multiples of each other
are non-parallel lines that do not intersect in 3D space
The of a line onto a plane can be found by projecting two points on the line onto the plane
Distance between point and line
Formula calculates the shortest distance d from a point P(x0,y0,z0) to a line defined by a position vector r1 and direction vector v
Numerator computes the magnitude of the cross product between the vector from r1 to P and the direction vector v
Denominator is the magnitude of the direction vector v
Cross product × finds the perpendicular vector between (r0−r1) and v, and its magnitude divided by ∣v∣ gives the perpendicular distance
Vector and scalar plane equations
Vector equation of a plane expresses the plane using a n perpendicular to the plane, a position vector r for any point on the plane, and a specific point r0 on the plane
⋅ between n and (r−r0) equals zero for all points on the plane
Example: ⟨1,2,3⟩⋅(r−⟨4,5,6⟩)=0
of a plane describes the plane using coefficients a, b, c from a normal vector and a constant term d determined by a point on the plane
Example: 1x+2y+3z+4=0
form a line where they meet, which can be determined using techniques
Point-to-plane distance in 3D
Formula finds the shortest distance d from a point P(x0,y0,z0) to a plane ax+by+cz+d=0
Numerator evaluates the left side of the at point P and takes the absolute value
Denominator is the magnitude of the normal vector ⟨a,b,c⟩
Ratio gives the perpendicular distance from the point to the plane
Angles Between Planes
Angle between intersecting planes
Formula determines the angle θ between two intersecting planes with normal vectors n1 and n2
Dot product n1⋅n2 in the numerator finds the projection of one normal vector onto the other
Denominator multiplies the magnitudes of the two normal vectors
Inverse cosine of the ratio gives the angle θ between the planes
Absolute values ensure the angle is between 0 and 90 degrees
Key Terms to Review (17)
Cartesian Coordinates: Cartesian coordinates are a system used to locate points in space by specifying their positions along orthogonal (perpendicular) axes. This coordinate system provides a way to describe the location of objects in two-dimensional (2D) or three-dimensional (3D) space using numerical values.
Direction Vector: A direction vector is a vector that indicates the direction of a line or a curve in space. It provides information about the orientation and slope of a parametric equation or the normal vector of a plane.
Distance Formula: The distance formula is a mathematical equation used to calculate the straight-line distance between two points in a three-dimensional space. It extends the concept of the Pythagorean theorem into three dimensions, taking into account the coordinates of each point. This formula is essential for understanding relationships between points, vectors, and geometric shapes in three-dimensional space, particularly in relation to lines and planes.
Dot Product: The dot product, also known as the scalar product, is a fundamental operation in linear algebra that combines two vectors to produce a scalar value. It is a crucial concept in various areas of mathematics, including vector analysis, physics, and computer science.
Intersecting Planes: Intersecting planes are two or more planes in three-dimensional space that share a common line of intersection. This line represents the set of all points that belong to both planes simultaneously, creating a unique intersection between the planes.
Linear Algebra: Linear algebra is a branch of mathematics that deals with the study of linear equations, vectors, matrices, and linear transformations. It provides a framework for understanding and solving problems that involve linear relationships between variables.
Normal Vector: A normal vector is a vector that is perpendicular or orthogonal to a given surface, curve, or plane in three-dimensional space. It is a fundamental concept in calculus, geometry, and physics, as it helps describe the orientation and properties of various geometric objects.
Orthogonal Projection: Orthogonal projection is the process of finding the closest point on a line or plane to a given point in space. It involves finding the perpendicular distance between the point and the line or plane, and projecting the point onto the line or plane along this perpendicular direction.
Parallel Lines: Parallel lines are two or more lines in a plane that never intersect, maintaining a constant distance between them. This concept is crucial in the study of equations of lines and planes in three-dimensional space.
Parametric Equation: A parametric equation is a set of equations that define a curve or surface in a coordinate system by expressing the coordinates as functions of one or more parameters. These equations allow for the representation of complex shapes and paths that cannot be easily described by a single equation in the standard Cartesian coordinate system.
Plane Equation: The plane equation is a mathematical representation that describes the equation of a plane in three-dimensional space. It defines the set of all points that lie on a particular plane, allowing for the visualization and analysis of planar surfaces within a three-dimensional coordinate system.
Position Vector: The position vector is a vector that represents the location of a point in space relative to a reference point or origin. It is a fundamental concept in the study of geometry, vector analysis, and various fields of mathematics and physics.
Scalar Equation: A scalar equation is a mathematical expression that represents a relationship between scalar quantities, which are values that have magnitude but no direction. In the context of 2.5 Equations of Lines and Planes in Space, scalar equations are used to describe the equations of lines and planes in three-dimensional space.
Skew Lines: Skew lines are a pair of lines in three-dimensional space that do not intersect and are not parallel. They are distinct from intersecting lines and parallel lines, as they maintain a constant distance between them throughout their length.
Symmetric equation: A symmetric equation represents a line or a plane in three-dimensional space by eliminating the parameter from its parametric equations, showing the relationship between the coordinates directly. This form is useful because it simplifies understanding and visualizing geometric relationships without needing to reference a parameter like 't'. Symmetric equations help in analyzing intersections and orientations of lines and planes in space.
Vector: A vector is a mathematical quantity that has both magnitude and direction. It is used to represent physical quantities such as force, velocity, and displacement, which require both a size and a direction to be fully described.
Vector Equation: A vector equation is a mathematical expression that describes the relationship between a vector and its components or other vectors. It is a powerful tool used to represent and analyze geometric objects, such as lines and planes, in three-dimensional space.