∞Calculus IV Unit 1 – Vectors and Vector–Valued Functions

Key Concepts and Definitions

• Vectors quantities with both magnitude and direction (velocity, force, displacement)
• Scalar quantities with only magnitude, no direction (speed, mass, time)
• Vector components $x$, $y$, and $z$ values that describe a vector in terms of the standard unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$
• Vector magnitude $\|\vec{v}\| = \sqrt{v_x^2 + v_y^2 + v_z^2}$ length of the vector
  • Calculated using the Pythagorean theorem in 2D or 3D space
• Unit vectors vectors with a magnitude of 1 that point in a specific direction
  • Standard unit vectors: $\hat{i}$ (1, 0, 0), $\hat{j}$ (0, 1, 0), and $\hat{k}$ (0, 0, 1)
• Vector equality two vectors are equal if and only if their corresponding components are equal
• Zero vector $\vec{0}$ has a magnitude of 0 and no specific direction

Vector Operations and Properties

• Vector addition $\vec{u} + \vec{v} = (u_x + v_x, u_y + v_y, u_z + v_z)$ component-wise addition
  • Geometrically, vector addition follows the parallelogram law or the triangle law
• Vector subtraction $\vec{u} - \vec{v} = (u_x - v_x, u_y - v_y, u_z - v_z)$ component-wise subtraction
• Scalar multiplication multiplying a vector by a scalar changes its magnitude but not its direction
  • $c\vec{v} = (cv_x, cv_y, cv_z)$, where $c$ is a scalar
• Dot product (scalar product) $\vec{u} \cdot \vec{v} = u_xv_x + u_yv_y + u_zv_z$ results in a scalar value
  • Geometrically, $\vec{u} \cdot \vec{v} = \|\vec{u}\| \|\vec{v}\| \cos{\theta}$, where $\theta$ is the angle between the vectors
• Cross product (vector product) $\vec{u} \times \vec{v} = (u_yv_z - u_zv_y, u_zv_x - u_xv_z, u_xv_y - u_yv_x)$ results in a vector perpendicular to both $\vec{u}$ and $\vec{v}$
  • Magnitude of the cross product: $\|\vec{u} \times \vec{v}\| = \|\vec{u}\| \|\vec{v}\| \sin{\theta}$
• Scalar triple product $\vec{u} \cdot (\vec{v} \times \vec{w})$ determines the volume of a parallelepiped formed by the three vectors

Vector-Valued Functions

• Vector-valued functions functions that assign a vector to each point in their domain
  • Example: $\vec{r}(t) = (x(t), y(t), z(t))$, where $t$ is usually time
• Limit of a vector-valued function $\lim_{t \to a} \vec{r}(t) = (\lim_{t \to a} x(t), \lim_{t \to a} y(t), \lim_{t \to a} z(t))$ component-wise limit
• Continuity of a vector-valued function $\vec{r}(t)$ is continuous if and only if its component functions $x(t)$, $y(t)$, and $z(t)$ are continuous
• Parametric curves curves in 2D or 3D space defined by a vector-valued function
  • Example: $\vec{r}(t) = (t^2, t^3)$ defines a parametric curve in the $xy$-plane
• Arc length $L = \int_a^b \|\vec{r}'(t)\| dt$ measures the length of a parametric curve over the interval $[a, b]$

Derivatives of Vector-Valued Functions

• Derivative of a vector-valued function $\vec{r}'(t) = (x'(t), y'(t), z'(t))$ component-wise differentiation
  • Geometrically, the derivative represents the tangent vector to the curve at a given point
• Second derivative $\vec{r}''(t) = (x''(t), y''(t), z''(t))$ component-wise second derivative
• Tangent line to a curve $\vec{r}(t)$ at $t = t_0$ is given by $\vec{l}(t) = \vec{r}(t_0) + (t - t_0)\vec{r}'(t_0)$
• Normal vector $\vec{N}(t)$ vector perpendicular to the tangent vector at a given point
  • Unit normal vector: $\hat{N}(t) = \frac{\vec{r}'(t)}{\|\vec{r}'(t)\|}$
• Binormal vector $\vec{B}(t) = \vec{T}(t) \times \vec{N}(t)$ vector perpendicular to both the tangent and normal vectors
• Curvature $\kappa(t) = \frac{\|\vec{r}'(t) \times \vec{r}''(t)\|}{\|\vec{r}'(t)\|^3}$ measures how much a curve deviates from a straight line

Integrals of Vector-Valued Functions

• Indefinite integral of a vector-valued function $\int \vec{r}(t) dt = (\int x(t) dt, \int y(t) dt, \int z(t) dt)$ component-wise integration
  • Constant of integration is a vector $\vec{C} = (C_x, C_y, C_z)$
• Definite integral of a vector-valued function $\int_a^b \vec{r}(t) dt = (\int_a^b x(t) dt, \int_a^b y(t) dt, \int_a^b z(t) dt)$ component-wise definite integration
• Average value of a vector-valued function $\frac{1}{b-a} \int_a^b \vec{r}(t) dt$ over the interval $[a, b]$
• Center of mass $\vec{r}_{cm} = \frac{\int_a^b \vec{r}(t) dm}{\int_a^b dm}$ for a system of particles or a continuous object
  • $dm$ represents the mass element, which can be a discrete particle mass or a continuous mass distribution

Applications in Physics and Engineering

• Position, velocity, and acceleration $\vec{r}(t)$, $\vec{v}(t) = \vec{r}'(t)$, and $\vec{a}(t) = \vec{v}'(t) = \vec{r}''(t)$
  • Projectile motion: $\vec{r}(t) = (v_0 \cos{\theta} t, v_0 \sin{\theta} t - \frac{1}{2}gt^2)$, where $v_0$ is the initial velocity, $\theta$ is the launch angle, and $g$ is the acceleration due to gravity
• Force $\vec{F} = m\vec{a}$ Newton's second law of motion
  • Work done by a force: $W = \int_C \vec{F} \cdot d\vec{r}$, where $C$ is the path along which the force acts
• Moment of a force (torque) $\vec{\tau} = \vec{r} \times \vec{F}$ measures the tendency of a force to cause rotation
• Fluid flow velocity fields $\vec{v}(x, y, z)$ describe the velocity of a fluid at each point in space
  • Streamlines, pathlines, and streaklines help visualize fluid flow patterns
• Electromagnetic fields electric field $\vec{E}(x, y, z)$ and magnetic field $\vec{B}(x, y, z)$ are vector fields that describe the forces acting on charged particles
  • Maxwell's equations govern the behavior of electromagnetic fields

Common Pitfalls and Misconceptions

• Confusing scalar and vector quantities speed (scalar) vs. velocity (vector), distance (scalar) vs. displacement (vector)
• Misunderstanding vector addition and subtraction geometrically or component-wise
  • Tip-to-tail method for vector addition and parallelogram law
• Misinterpreting dot product and cross product results dot product yields a scalar, cross product yields a vector
  • Sign of the dot product indicates acute ($> 0$), obtuse ($< 0$), or perpendicular ($= 0$) angles between vectors
• Forgetting to normalize vectors when finding unit tangent, normal, or binormal vectors
• Incorrectly applying the chain rule when differentiating vector-valued functions
  • $\frac{d}{dt} \vec{r}(t) = \vec{r}'(t) \frac{dt}{dt} = \vec{r}'(t)$, not $\frac{d}{dt} \vec{r}(t) = (\frac{d}{dt} x(t), \frac{d}{dt} y(t), \frac{d}{dt} z(t))$
• Misinterpreting the meaning of the integral of a vector-valued function
  • Indefinite integral yields a vector-valued function, definite integral yields a vector

Practice Problems and Solutions

1. Find the unit vector in the direction of $\vec{v} = (3, -4, 0)$.

   • Solution: $\hat{v} = \frac{\vec{v}}{\|\vec{v}\|} = (\frac{3}{5}, -\frac{4}{5}, 0)$

2. Determine the angle between the vectors $\vec{u} = (1, 2, -1)$ and $\vec{v} = (2, 0, 3)$.

   • Solution: $\cos{\theta} = \frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\| \|\vec{v}\|} = \frac{5}{\sqrt{6} \sqrt{13}}$, so $\theta \approx 49.9°$

3. Find the area of the parallelogram formed by the vectors $\vec{a} = (1, 2)$ and $\vec{b} = (-3, 1)$.

   • Solution: Area $= \|\vec{a} \times \vec{b}\| = \|(-2, -7)\| = \sqrt{53}$

4. Given the vector-valued function $\vec{r}(t) = (t^2, \sin{t}, e^t)$, find $\vec{r}'(t)$ and $\vec{r}''(t)$.

   • Solution: $\vec{r}'(t) = (2t, \cos{t}, e^t)$ and $\vec{r}''(t) = (2, -\sin{t}, e^t)$

5. Evaluate the definite integral $\int_0^1 (3t^2, 2t, 1) dt$.

   • Solution: $\int_0^1 (3t^2, 2t, 1) dt = (1, 1, 1)$

6. A particle moves along the curve $\vec{r}(t) = (2\cos{t}, 2\sin{t}, t)$. Find its speed at $t = \frac{\pi}{4}$.

   • Solution: Speed $= \|\vec{r}'(t)\| = \sqrt{(-2\sin{t})^2 + (2\cos{t})^2 + 1^2} = \sqrt{5}$ at $t = \frac{\pi}{4}$

7. Find the work done by the force $\vec{F}(x, y) = (x^2, y)$ along the path $\vec{r}(t) = (t, t^2)$ from $t = 0$ to $t = 1$.

   • Solution: $W = \int_C \vec{F} \cdot d\vec{r} = \int_0^1 (t^2, t^2) \cdot (1, 2t) dt = \int_0^1 (t^2 + 2t^3) dt = \frac{7}{12}$

8. Determine the curvature of the helix $\vec{r}(t) = (\cos{t}, \sin{t}, t)$ at $t = \frac{\pi}{2}$.

   • Solution: $\kappa(t) = \frac{\|\vec{r}'(t) \times \vec{r}''(t)\|}{\|\vec{r}'(t)\|^3} = \frac{1}{\sqrt{2}}$ at $t = \frac{\pi}{2}$