Topic 12: Vectors & Three-Dimensional Geometry
Welcome to this introductory section which expands upon our understanding of space by extending geometric and algebraic concepts into the three-dimensional realm. This topic introduces vectors as fundamental and powerful tools for handling quantities that possess not just magnitude, but also a specific direction. Three-Dimensional (3D) Geometry provides the essential framework necessary for accurately describing the position, shape, size, and orientation of objects within the space that surrounds us. It builds upon 2D geometry by adding depth as a third dimension.
Our starting point in 3D Geometry is the establishment of the 3D Cartesian coordinate system. This system uses three mutually perpendicular axes, conventionally labeled as the x, y, and z axes, intersecting at an origin (0,0,0). We learn to define the coordinates of any point in space using an ordered triplet $(x, y, z)$ and extend familiar concepts from 2D geometry, such as the distance formula between two points, to their 3D equivalents. A key element in describing the orientation of lines in space is understanding direction cosines (the cosines of the angles a line makes with the positive x, y, and z axes) and direction ratios (any set of numbers proportional to the direction cosines).
The introduction of vectors marks a significant shift in perspective. Vectors are conceptualized as directed line segments, serving as mathematical representations for physical quantities that inherently possess both a specific magnitude and a direction. Examples of such quantities include displacement, velocity, force, and acceleration. We delve into vector algebra, learning how to perform fundamental operations: vector addition (visualized using the parallelogram law or triangle law), vector subtraction, and scalar multiplication (which changes the magnitude of a vector). The concept of position vectors, which locate points in space relative to a chosen origin, is also introduced. Different types of vectors, such as the zero vector, unit vector (magnitude 1), and collinear vectors, are defined based on their properties.
Two critically important operations involving vectors are the scalar product and the vector product. The scalar (or dot) product of two vectors, $\mathbf{A}$ and $\mathbf{B}$ (denoted $\mathbf{A} \cdot \mathbf{B}$), yields a scalar value. This value is related to the projection of one vector onto the other and is particularly useful for finding the angle $\theta$ between two vectors using the formula $\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos \theta$. It also provides a simple test for perpendicularity: if $\mathbf{A} \cdot \mathbf{B} = 0$ (and $\mathbf{A}, \mathbf{B}$ are non-zero), then $\mathbf{A}$ and $\mathbf{B}$ are perpendicular.
The vector (or cross) product of $\mathbf{A}$ and $\mathbf{B}$ (denoted $\mathbf{A} \times \mathbf{B}$) yields a new vector. This resulting vector is perpendicular to the plane containing the original two vectors $\mathbf{A}$ and $\mathbf{B}$. Its magnitude is given by $|\mathbf{A} \times \mathbf{B}| = |\mathbf{A}||\mathbf{B}|\sin \theta$, where $\theta$ is the angle between $\mathbf{A}$ and $\mathbf{B}$. The direction is given by the right-hand rule (represented by the unit vector $\hat{\mathbf{n}}$ in $\mathbf{A} \times \mathbf{B} = |\mathbf{A}||\mathbf{B}|\sin \theta \, \hat{\mathbf{n}}$). The cross product is invaluable for calculating areas of parallelograms or triangles defined by two vectors, and it has significant applications in physics, such as determining torque or magnetic force.
This vector-based framework provides powerful tools for describing geometric entities in 3D space. We use vectors to derive and represent the equations of lines in space, typically presented in both vector form and Cartesian form. Similarly, we derive the equations of planes in space, exploring various representations such as vector form, normal form, intercept form, and general Cartesian form. With these equations established, we can then analyze the complex relationships between lines and planes. This includes finding the angles between lines, between planes, or between a line and a plane; determining conditions under which they intersect, are parallel, or are perpendicular; calculating the shortest distance between two skew lines (lines that are neither parallel nor intersecting) or the distance from a point to a line or plane. An introduction to the equation of a sphere might also be included as part of this topic's scope.
The concepts covered in this topic are absolutely fundamental and have widespread applications across numerous fields. They are essential for understanding problems in physics (especially mechanics, motion, and electromagnetism), various branches of engineering, computer graphics (for rendering and transformations), robotics (for movement and manipulation), and navigation. The ability to model and analyze spatial relationships using 3D geometry and vectors is a critical skill in quantitative disciplines.
Vector Algebra: Introduction and Basic Operations
Vector algebra introduces quantities possessing both magnitude and direction, distinct from scalars which only have magnitude. Key concepts include defining a vector, its magnitude (length), and direction. Various types of vectors exist, such as zero vectors, unit vectors (magnitude 1), coinitial vectors (same starting point), collinear vectors (parallel to the same line), and equal vectors (same magnitude and direction). Basic operations like vector addition (using triangle or parallelogram law), subtraction, and multiplication by a scalar are fundamental. These operations follow specific properties, such as commutativity and associativity for addition, and distributivity for scalar multiplication.
Components of a Vector and Linear Combinations
Understanding vectors using their components provides a powerful analytical tool. A position vector $\vec{OP}$ of a point $P(x, y, z)$ relative to the origin $O(0, 0, 0)$ is given by $x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}$. A vector joining two points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ is $\vec{AB} = (x_2-x_1)\hat{\mathbf{i}} + (y_2-y_1)\hat{\mathbf{j}} + (z_2-z_1)\hat{\mathbf{k}}$. The magnitude of a vector $\mathbf{v} = a\hat{\mathbf{i}} + b\hat{\mathbf{j}} + c\hat{\mathbf{k}}$ is $|\mathbf{v}| = \sqrt{a^2 + b^2 + c^2}$. Vectors can be expressed as a linear combination of basis vectors, like $\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}$. Components simplify checking collinearity ($\mathbf{a} = \lambda \mathbf{b}$) and coplanarity.
Scalar (Dot) Product of Vectors
The scalar product, or dot product, of two vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta$, where $\theta$ is the angle between them. The result is a scalar. Key properties include commutativity ($\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$) and distributivity over addition. In terms of components, if $\mathbf{a} = a_1\hat{\mathbf{i}} + a_2\hat{\mathbf{j}} + a_3\hat{\mathbf{k}}$ and $\mathbf{b} = b_1\hat{\mathbf{i}} + b_2\hat{\mathbf{j}} + b_3\hat{\mathbf{k}}$, then $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$. Geometrically, the dot product relates to the projection of one vector onto another. Applications include finding the angle between vectors ($\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}$) and calculating the work done by a force ($\mathbf{W} = \mathbf{F} \cdot \mathbf{d}$).
Vector (Cross) Product of Vectors
The vector product, or cross product, of two vectors $\mathbf{a}$ and $\mathbf{b}$ is a vector defined as $\mathbf{a} \times \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \sin \theta \hat{\mathbf{n}}$, where $\theta$ is the angle between them and $\hat{\mathbf{n}}$ is a unit vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$, following the right-hand rule. The result is a vector. Unlike the dot product, the cross product is non-commutative ($\mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a}$). In components, it's calculated using a determinant: $\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$. Geometrically, the magnitude $|\mathbf{a} \times \mathbf{b}|$ equals the area of the parallelogram formed by $\mathbf{a}$ and $\mathbf{b}$. Applications include finding a vector normal to a plane and calculating torque.
Scalar Triple Product
The scalar triple product of three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ is defined as $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$. The result is a scalar. It can be calculated using the determinant of the matrix formed by the component vectors: $[\mathbf{a} \, \mathbf{b} \, \mathbf{c}] = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$. A key property is cyclic permutation: $[\mathbf{a} \, \mathbf{b} \, \mathbf{c}] = [\mathbf{b} \, \mathbf{c} \, \mathbf{a}] = [\mathbf{c} \, \mathbf{a} \, \mathbf{b}]$. Geometrically, the magnitude of the scalar triple product $| \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) |$ represents the volume of the parallelepiped formed by the three vectors. The most significant application is determining coplanarity: three vectors are coplanar if and only if their scalar triple product is zero.
Section Formula in Vector Algebra
The section formula in vector algebra determines the position vector of a point that divides a line segment joining two points with given position vectors. If points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$, and a point $R$ divides the segment $AB$ internally in the ratio $m:n$, its position vector $\mathbf{r}$ is given by $\mathbf{r} = \frac{n\mathbf{a} + m\mathbf{b}}{m+n}$. For external division in the ratio $m:n$, the formula is $\mathbf{r} = \frac{n\mathbf{a} - m\mathbf{b}}{n-m}$. A special case is the midpoint formula, where $m:n = 1:1$, resulting in $\mathbf{r} = \frac{\mathbf{a} + \mathbf{b}}{2}$. This formula can be implicitly extended to find the centroid of a triangle or tetrahedron.
Three Dimensional Geometry: Introduction and Lines
Three-dimensional geometry extends coordinate systems to three axes ($x, y, z$), allowing points in space to be represented as ordered triplets $(x, y, z)$. Fundamental concepts include finding the distance between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$, using the distance formula $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$, and the section formula for points dividing a line segment in 3D. Direction cosines $(l, m, n)$ and direction ratios $(a, b, c)$ describe the direction of a line, with $l^2+m^2+n^2=1$. The equation of a line in space can be expressed in vector form $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$ or Cartesian form $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$, with simple conversion between the two.
Three Dimensional Geometry: Planes
Planes are fundamental surfaces in three-dimensional geometry. Their equations can be represented in various forms. The vector form can be normal form ($\mathbf{r} \cdot \hat{\mathbf{n}} = d$) or point-normal form ($\mathbf{r} - \mathbf{a} \cdot \hat{\mathbf{n}} = 0$). The Cartesian form includes the general linear equation $Ax + By + Cz + D = 0$, the normal form $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ (intercept form), and the form derived from three non-collinear points. Understanding these different representations allows for analysis of planar surfaces, including their orientation and position relative to the origin and other geometric entities.
Three Dimensional Geometry: Angles
Calculating angles between geometric entities is crucial in 3D geometry. The angle between two lines is found using the dot product of their direction vectors: $\cos \theta = \frac{|\mathbf{b}_1 \cdot \mathbf{b}_2|}{|\mathbf{b}_1| |\mathbf{b}_2|}$. The angle between two planes is determined by the angle between their normal vectors: $\cos \theta = \frac{|\mathbf{n}_1 \cdot \mathbf{n}_2|}{|\mathbf{n}_1| |\mathbf{n}_2|}$. The angle $\phi$ between a line and a plane is complementary to the angle $\theta$ between the line's direction vector $\mathbf{b}$ and the plane's normal vector $\mathbf{n}$: $\sin \phi = \frac{|\mathbf{b} \cdot \mathbf{n}|}{|\mathbf{b}| |\mathbf{n}|}$. Conditions for parallelism and perpendicularity are derived directly from these angular relationships.
Three Dimensional Geometry: Distances
Finding distances between points, lines, and planes is a core aspect of 3D geometry. The shortest distance between two skew lines (non-parallel, non-intersecting) can be calculated using a formula involving the scalar triple product or vector methods. The distance between parallel lines is found by taking a point on one line and calculating its distance to the other line. The distance of a point $P_0(x_0, y_0, z_0)$ from a plane $Ax + By + Cz + D = 0$ is given by $d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}$. Methods also exist for finding the distance of a point from a line in 3D space.