Vector Calculus and Analytic Geometry: Study Guide with Solutions

Vector Calculus and Analytic Geometry

Geometric Descriptions of Sets of Points

This section focuses on describing sets of points in space using equations and geometric terminology.

• Circle in a Plane: The equation $x^2 + y^2 = 48$, $z = 1$ describes a circle of radius $oxed{ ext{radius} = ext{sqrt}(48)}$ lying in the plane $z = 1$.

• Sphere: The equation $x^2 + y^2 + z^2 = 9$ describes a sphere of radius $3$centered at the origin. Points outside the sphere satisfy$x^2 + y^2 + z^2 > 9$.

• Region Between Planes: The inequalities $10 eq z eq 11$ describe the region between the planes $z = 10$ and $z = 11$.

Example: The set $x^2 + y^2 = 48$, $z = 1$ is a circle of radius $oxed{4 ext{sqrt}(3)}$ in the plane $z = 1$.

Equations of Planes and Spheres

Planes and spheres are fundamental surfaces in three-dimensional analytic geometry.

• Plane Equation: A plane passing through point $(-1, -4, 5)$ and parallel to the $yz$-plane has the equation $x = -1$.

• Sphere Equation: A sphere centered at $(0, -2, 2)$ with radius $5$has the equation$(x - 0)^2 + (y + 2)^2 + (z - 2)^2 = 25$.

Example: The plane $x = -1$ passes through $(-1, -4, 5)$ and is parallel to the $yz$-plane.

Distance Between Points

The distance between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ in space is given by:

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Example: The distance between $(8, 5, -1)$ and $(2, 6, -2)$ is $\sqrt{(8-2)^2 + (5-6)^2 + (-1+2)^2} = \sqrt{36 + 1 + 1} = \sqrt{38}$.

Vectors: Representation, Length, and Direction

Vectors are quantities with both magnitude and direction, represented in component form.

• Vector from Two Points: The vector from $A$ to $B$ is $\vec{AB} = (x_B - x_A)\mathbf{i} + (y_B - y_A)\mathbf{j} + (z_B - z_A)\mathbf{k}$.

• Length (Magnitude): $|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$

• Direction: The unit vector in the direction of $\vec{v}$ is $\frac{\vec{v}}{|\vec{v}|}$.

Example: For $\vec{v} = 5\mathbf{i} - 8\mathbf{j} + 5\mathbf{k}$, $|\vec{v}| = \sqrt{5^2 + (-8)^2 + 5^2} = \sqrt{25 + 64 + 25} = \sqrt{114}$.

Vector Operations: Addition, Scalar Multiplication, Dot and Cross Product

Vectors can be added, scaled, and combined using dot and cross products.

• Addition: $\vec{u} + \vec{v} = (u_x + v_x)\mathbf{i} + (u_y + v_y)\mathbf{j} + (u_z + v_z)\mathbf{k}$

• Scalar Multiplication: $c\vec{v} = (cv_x)\mathbf{i} + (cv_y)\mathbf{j} + (cv_z)\mathbf{k}$

• Dot Product: $\vec{u} \cdot \vec{v} = u_x v_x + u_y v_y + u_z v_z$

• Cross Product: $\vec{u} \times \vec{v} = (u_y v_z - u_z v_y)\mathbf{i} - (u_x v_z - u_z v_x)\mathbf{j} + (u_x v_y - u_y v_x)\mathbf{k}$

Example: $\vec{u} = 13\mathbf{i} - 7\mathbf{j} - 5\mathbf{k}$, $\vec{v} = 3\mathbf{i} - 9\mathbf{j}$, $\vec{u} + \vec{v} = (13+3)\mathbf{i} + (-7-9)\mathbf{j} + (-5+0)\mathbf{k} = 16\mathbf{i} - 16\mathbf{j} - 5\mathbf{k}$.

Projections and Components

The projection of one vector onto another is a measure of how much one vector extends in the direction of another.

• Projection Formula: $\text{proj}_{\vec{u}} \vec{v} = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}|^2} \vec{u}$

Example: Project $\vec{v}$ onto $\vec{u}$ where $\vec{v} = 7\mathbf{i} - \mathbf{j} + 4\mathbf{k}$, $\vec{u} = 3\mathbf{i} - 9\mathbf{j}$.

Equations of Lines and Planes

Lines and planes in space can be described using vector and parametric equations.

• Line through Point and Parallel to Vector: $\vec{r}(t) = \vec{r}_0 + t\vec{v}$

• Plane Equation: $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$ where $(a, b, c)$ is the normal vector.

Example: The line through $(1, 2, 4)$ parallel to $2\mathbf{i} + 5\mathbf{j}$: $x = 1 + 2t$, $y = 2 + 5t$, $z = 4$.

Area and Volume Calculations

Areas and volumes in vector calculus are often computed using cross products and determinants.

• Area of Triangle: $A = \frac{1}{2}|\vec{AB} \times \vec{AC}|$

• Volume of Parallelepiped: $V = |\vec{a} \cdot (\vec{b} \times \vec{c})|$

Example: The area of a triangle with vertices $P_1, P_2, P_3$ is $\frac{1}{2}|\vec{P_2P_1} \times \vec{P_3P_1}|$.

Triple Scalar Product

The triple scalar product gives the volume of a parallelepiped defined by three vectors.

• Formula: $V = \vec{a} \cdot (\vec{b} \times \vec{c})$

Example: For $\vec{a} = 5\mathbf{i} - 2\mathbf{j} + 3\mathbf{k}$, $\vec{b} = 6\mathbf{k} - 7\mathbf{j} - 5\mathbf{i}$, $\vec{c} = 7\mathbf{i} - 5\mathbf{j}$, compute $V$.

Parametric Equations for Curves and Surfaces

Parametric equations describe curves and surfaces using one or more parameters.

• Line Segment: $\vec{r}(t) = (1-t)\vec{A} + t\vec{B}$ for $0 \leq t \leq 1$

• Plane: $ax + by + cz = d$

Example: The line segment joining $(2, 7, 42)$ and $(2, 0, 47)$: $x = 2$, $y = 7 - 7t$, $z = 42 + 5t$, $0 \leq t \leq 1$.

Intersection and Distance to Planes and Lines

Finding intersections and distances involves solving systems of equations and using projection formulas.

• Intersection: Solve the system of equations for the point(s) of intersection.

• Distance from Point to Line: $d = \frac{|(\vec{AP} \times \vec{v})|}{|\vec{v}|}$

Example: The distance from $(5, 1, -4)$ to the line $x = -2 + 2t$, $y = -1 + 6t$, $z = 6 + 9t$ is $\frac{\sqrt{111}}{3}$.

Vector Functions and Motion

Vector functions describe the position, velocity, and acceleration of particles in space.

• Position Vector: $\vec{r}(t)$

• Velocity: $\vec{v}(t) = \frac{d\vec{r}}{dt}$

• Acceleration: $\vec{a}(t) = \frac{d^2\vec{r}}{dt^2}$

Example: If $\vec{r}(t) = (10t)\mathbf{i} - (6t^2)\mathbf{j} + (5t^3)\mathbf{k}$, then $\vec{v}(t) = 10\mathbf{i} - 12t\mathbf{j} + 15t^2\mathbf{k}$.

Projectile Motion

Projectile motion problems involve analyzing the path of an object under gravity.

• Horizontal Launch: $y = y_0 + v_0 t - \frac{1}{2}gt^2$

• Time of Flight: Solve for $t$ when $y = 0$.

Example: A projectile is fired at $720$m/s at $45^\circ$; time to hit the ground is $104$ s.

Arc Length of a Curve

The arc length of a curve $\vec{r}(t)$ from $t=a$ to $t=b$ is:

• Arc Length Formula: $L = \int_a^b |\vec{r}'(t)| dt$

Example: For $\vec{r}(t) = (6t)\mathbf{i} + (9t^2)\mathbf{j} + (2t^3)\mathbf{k}$, $L = \int_0^1 \sqrt{36 + 324t^2 + 36t^4} dt$.

Unit Tangent and Normal Vectors

The unit tangent vector $\mathbf{T}$ and principal unit normal vector $\mathbf{N}$ describe the direction and curvature of a curve.

• Unit Tangent: $\mathbf{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|}$

• Principal Unit Normal: $\mathbf{N}(t) = \frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|}$

Example: For $\vec{r}(t) = (10t)\mathbf{i} + (2\sin t)\mathbf{j} + (10\cos t)\mathbf{k}$, $\mathbf{T}(t) = \frac{1}{\sqrt{104}}(10\mathbf{i} + 2\cos t\mathbf{j} - 10\sin t\mathbf{k})$.

Curvature of Space Curves

Curvature measures how sharply a curve bends at a given point.

• Curvature Formula: $\kappa = \frac{|\vec{r}'(t) \times \vec{r}''(t)|}{|\vec{r}'(t)|^3}$

Example: For $\vec{r}(t) = (2t)\mathbf{i} + (t^2)\mathbf{j}$, $\kappa = \frac{2}{(4t^2 + 1)^{3/2}}$.

Integrals and Initial Value Problems

Definite integrals and initial value problems are used to solve for quantities and functions in calculus.

• Definite Integral: $\int_a^b f(t) dt$

• Initial Value Problem: Solve $\frac{d\vec{r}}{dt} = \vec{v}(t)$ with $\vec{r}(0) = \vec{r}_0$.

Example: $\int_0^1 [2e^{2t} - 3\sin t - 2\cos t] dt$

Summary Table: Key Vector Operations

Operation	Formula	Geometric Meaning
Dot Product	$\vec{a} \cdot \vec{b}$	Projection, angle between vectors
Cross Product	$\vec{a} \times \vec{b}$	Area of parallelogram, perpendicular vector
Magnitude	$|\vec{a}|$	Length of vector
Unit Vector	$\frac{\vec{a}}{|\vec{a}|}$	Direction only
Projection	$\text{proj}_{\vec{a}} \vec{b}$	Component of $\vec{b}$ along $\vec{a}$

Additional info:

• Some problems involve parametric equations, projectile motion, and curvature, which are topics in multivariable calculus and analytic geometry.

• All equations are provided in LaTeX format for clarity and academic rigor.