5️⃣Multivariable Calculus Unit 1 – Vectors and the Geometry of Space

Key Concepts and Definitions

• Vectors quantities have both magnitude and direction, while scalars only have magnitude
• Magnitude measures the length or size of a vector, denoted as $\lVert \vec{v} \rVert$
• Direction specifies the orientation of a vector in space, often given as an angle or unit vector
  • Unit vectors have a magnitude of 1 and indicate a specific direction (i, j, k)
• Dot product of two vectors results in a scalar value, calculated as $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$
  • Geometrically, the dot product is related to the angle between two vectors: $\vec{a} \cdot \vec{b} = \lVert \vec{a} \rVert \lVert \vec{b} \rVert \cos \theta$
• Cross product of two vectors results in a new vector perpendicular to both original vectors, calculated as $\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)$
  • The magnitude of the cross product is equal to the area of the parallelogram formed by the two vectors: $\lVert \vec{a} \times \vec{b} \rVert = \lVert \vec{a} \rVert \lVert \vec{b} \rVert \sin \theta$
• Vector-valued functions map real numbers to vectors, representing curves or paths in space

Vector Basics and Operations

• Vectors can be represented using various notations, such as $\vec{v}$, $\mathbf{v}$, or $\overrightarrow{AB}$
• Vector addition follows the parallelogram rule or triangle rule, resulting in a new vector from the tail of one vector to the head of the other
• Scalar multiplication scales a vector by a real number, changing its magnitude but not its direction: $c\vec{v} = (cv_1, cv_2, cv_3)$
• Vector subtraction is defined as adding the negative of a vector: $\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$
• Vectors can be described using components (x, y, z) or in terms of magnitude and direction angles
  • Converting between rectangular and spherical or cylindrical coordinates is often necessary
• Linear combinations of vectors, such as $c_1\vec{v_1} + c_2\vec{v_2}$, play a crucial role in many applications
• Orthogonal vectors have a dot product of zero, meaning they are perpendicular to each other

Three-Dimensional Coordinate Systems

• Cartesian (rectangular) coordinates (x, y, z) are the most common 3D coordinate system
  • Each axis is perpendicular to the others, forming a right-handed system
• Cylindrical coordinates $(r, \theta, z)$ are useful for problems with circular symmetry
  • $r$ is the distance from the z-axis, $\theta$ is the angle in the xy-plane, and $z$ is the height
• Spherical coordinates $(\rho, \theta, \phi)$ are advantageous for problems with spherical symmetry
  • $\rho$ is the distance from the origin, $\theta$ is the azimuthal angle in the xy-plane, and $\phi$ is the polar angle from the z-axis
• Converting between coordinate systems involves trigonometric functions and vector operations
  • Example: $(x, y, z) = (r\cos\theta, r\sin\theta, z)$ converts cylindrical to Cartesian coordinates
• Understanding the relationships between coordinate systems is crucial for solving problems in 3D space

Lines and Planes in Space

• Lines in 3D can be represented using parametric equations, vector equations, or symmetric equations
  • Parametric: $x = x_0 + at$, $y = y_0 + bt$, $z = z_0 + ct$, where $(x_0, y_0, z_0)$ is a point on the line and $(a, b, c)$ is a parallel vector
  • Vector: $\vec{r}(t) = \vec{r_0} + t\vec{v}$, where $\vec{r_0}$ is a position vector of a point on the line and $\vec{v}$ is a direction vector
• Planes can be described using a point and a normal vector or by a linear equation in x, y, and z
  • Normal vector form: $\vec{n} \cdot (\vec{r} - \vec{r_0}) = 0$, where $\vec{n}$ is the normal vector and $\vec{r_0}$ is a position vector of a point on the plane
  • Linear equation: $ax + by + cz + d = 0$, where $(a, b, c)$ is a normal vector and $d$ is a constant
• The angle between two planes or a line and a plane can be found using dot products of their normal vectors
• The intersection of a line and a plane, or two planes, can be determined by solving systems of equations
  • Example: A line with parametric equations $x = 1 + 2t$, $y = -1 + 3t$, $z = 4 - t$ intersecting the plane $2x - y + z = 5$ results in the point $(2, 1, 3)$ at $t = 0.5$

Vector-Valued Functions

• Vector-valued functions assign a vector to each input value, often representing position, velocity, or acceleration in space
  • Example: $\vec{r}(t) = (cos(t), sin(t), t)$ describes a helix in 3D space
• Limits, derivatives, and integrals of vector-valued functions are computed component-wise
  • The derivative $\frac{d\vec{r}}{dt}$ represents the tangent vector or velocity at a given point
  • The integral $\int \vec{r}(t) dt$ can be used to find displacement or work done along a curve
• Arc length of a curve $\vec{r}(t)$ from $t=a$ to $t=b$ is calculated as $\int_a^b \lVert \vec{r}'(t) \rVert dt$
• Curvature measures how quickly a curve changes direction, given by $\kappa(t) = \frac{\lVert \vec{r}'(t) \times \vec{r}''(t) \rVert}{\lVert \vec{r}'(t) \rVert^3}$
  • Higher curvature indicates a more rapidly changing direction, while lower curvature suggests a straighter path
• Motion along a curve can be analyzed using vector-valued functions for position, velocity, and acceleration

Applications in Physics and Engineering

• Vectors are essential for modeling forces, velocities, and accelerations in physics
  • Newton's second law, $\vec{F} = m\vec{a}$, relates the net force to mass and acceleration vectors
  • Work done by a force along a path is the dot product of force and displacement vectors: $W = \vec{F} \cdot \vec{d}$
• Electromagnetic fields, such as electric and magnetic fields, are represented by vector fields in 3D space
  • The electric field due to a point charge is given by Coulomb's law: $\vec{E} = \frac{kq}{r^2}\hat{r}$, where $\hat{r}$ is the unit vector pointing from the charge to the field point
• Fluid dynamics uses vector fields to describe fluid velocity, pressure, and density at each point in space
  • The Navier-Stokes equations, a set of partial differential equations, model fluid flow using vector calculus concepts
• Robotics and computer graphics rely on vector operations for transformations, rotations, and translations in 3D space
  • Rotation matrices, composed of orthogonal unit vectors, are used to rotate objects or coordinate systems
• Structural analysis in engineering employs vectors to calculate forces, moments, and stresses on beams, trusses, and frames
  • Example: A force of (100, -50, 75) N acting at a point (2, 3, -1) m creates a moment vector of (-125, -325, -250) N·m

Common Challenges and Solutions

• Visualizing vectors and 3D geometry can be difficult; using graphical tools or physical models can help build intuition
  • Software like GeoGebra, MATLAB, or Mathematica can create interactive 3D plots and animations
• Keeping track of signs and components in vector calculations is crucial; organize work carefully and double-check results
  • Consistency in notation (e.g., always using i, j, k for unit vectors) can reduce errors
• Choosing the appropriate coordinate system for a problem can simplify calculations; consider symmetry and constraints
  • Example: Using cylindrical coordinates for a problem involving a cylinder aligned with the z-axis can lead to more straightforward equations
• Remembering vector identities and properties, such as the cross product's cyclic nature or the Jacobi identity, takes practice
  • Create a reference sheet with key formulas and properties, and apply them regularly in problem-solving
• Interpreting the physical meaning of vector operations and results is as important as the calculations themselves
  • Example: Recognizing that the dot product of velocity and acceleration vectors relates to the rate of change of speed can provide insight into a particle's motion

Practice Problems and Examples

• Find the angle between the vectors $\vec{a} = (1, 2, -3)$ and $\vec{b} = (4, -2, 1)$ using the dot product
  • Solution: $\cos \theta = \frac{\vec{a} \cdot \vec{b}}{\lVert \vec{a} \rVert \lVert \vec{b} \rVert} = \frac{1}{\sqrt{14}\sqrt{21}} \approx 0.1336$, so $\theta \approx 82.3°$
• Determine the point of intersection between the line $\frac{x-1}{2} = \frac{y+3}{-1} = \frac{z-2}{4}$ and the plane $3x - 2y + z = 7$
  • Solution: Parametric equations for the line are $x = 1 + 2t$, $y = -3 - t$, $z = 2 + 4t$. Substituting into the plane equation yields $t = 1$, so the intersection point is $(3, -4, 6)$
• Find the arc length of the curve $\vec{r}(t) = (e^t \cos t, e^t \sin t, e^t)$ from $t = 0$ to $t = \pi/2$
  • Solution: $\vec{r}'(t) = (e^t(\cos t - \sin t), e^t(\sin t + \cos t), e^t)$, so $\lVert \vec{r}'(t) \rVert = \sqrt{3}e^t$. Arc length $= \int_0^{\pi/2} \sqrt{3}e^t dt = \sqrt{3}(e^{\pi/2} - 1) \approx 4.27$
• A particle moves along the path $\vec{r}(t) = (t^2, t^3, t)$ from $t = 1$ to $t = 2$. Find the work done by the force field $\vec{F}(x, y, z) = (xy, yz, xz)$
  • Solution: $\vec{r}'(t) = (2t, 3t^2, 1)$, so $\vec{F}(\vec{r}(t)) = (t^5, t^4, t^3)$. Work $= \int_1^2 \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) dt = \int_1^2 (2t^6 + 3t^6 + t^3) dt = \frac{4801}{7} \approx 685.86$
• Verify Lagrange's identity for the vectors $\vec{a} = (1, -2, 3)$, $\vec{b} = (-4, 2, 1)$, and $\vec{c} = (2, 0, -1)$
  • Solution: $(\vec{a} \times \vec{b}) \cdot \vec{c} = (7, 13, 10) \cdot (2, 0, -1) = 4$, and $(\vec{a} \cdot \vec{c})(\vec{b} \cdot \vec{c}) - (\vec{a} \cdot \vec{b})(\vec{c} \cdot \vec{c}) = (1)(0) - (-5)(6) = 30$, so Lagrange's identity holds