Chapter 12: Vectors and the Geometry of Space

Section 1: Vectors in the Plane and Space

This section introduces the concept of vectors in two and three dimensions, including their properties and operations.

• Vector Definition: A vector is a quantity with both magnitude and direction, represented as an ordered pair or triple, e.g., \( \vec{v} = \langle v_1, v_2, v_3 \rangle \).

• Vector Operations: Vectors can be added, subtracted, and multiplied by scalars.

• Magnitude: The length of a vector \( \vec{v} = \langle v_1, v_2, v_3 \rangle \) is given by:

• Unit Vector: A vector of length 1 in the direction of \( \vec{v} \):

• Example: Find the magnitude and unit vector of \( \vec{a} = \langle 3, 4, 0 \rangle \).

Section 2: The Dot Product and Angle Between Vectors

The dot product is a way to multiply two vectors to obtain a scalar, useful for finding angles and projections.

• Dot Product Formula:

• Angle Between Vectors: where \( \theta \) is the angle between \( \vec{a} \) and \( \vec{b} \).

• Orthogonality: Vectors are orthogonal (perpendicular) if their dot product is zero.

• Example: Find the angle between \( \vec{a} = \langle 1, 2, 3 \rangle \) and \( \vec{b} = \langle 4, -5, 6 \rangle \).

Section 3: Projections and Decomposition

This section covers how to project one vector onto another and decompose a vector into components parallel and orthogonal to a given vector.

• Projection Formula: The projection of \( \vec{F} \) onto \( \vec{v} \):

• Decomposition: Any vector \( \vec{F} \) can be written as the sum of its projection onto \( \vec{v} \) and a vector orthogonal to \( \vec{v} \):

• Example: Decompose \( \vec{F} = \langle 2, 3, 4 \rangle \) into components parallel and orthogonal to \( \vec{v} = \langle 1, 0, 0 \rangle \).

Section 4: The Cross Product and Box Product

The cross product of two vectors in three dimensions results in a vector orthogonal to both.

• Cross Product Formula:

• Properties: The cross product is anti-commutative and distributive over addition.

• Box Product (Scalar Triple Product): This gives the volume of the parallelepiped formed by the three vectors.

• Example: Compute \( \vec{a} \times \vec{b} \) for \( \vec{a} = \langle 1, 2, 3 \rangle \), \( \vec{b} = \langle 4, 5, 6 \rangle \).

Section 5: Equations of Lines and Planes; Distances

This section discusses how to find equations for lines and planes in space, and how to compute distances.

• Equation of a Line: Through point \( P_0 = (x_0, y_0, z_0) \) in the direction of vector \( \vec{v} = \langle a, b, c \rangle \):

• Equation of a Plane: Through point \( P_0 = (x_0, y_0, z_0) \) with normal vector \( \vec{n} = \langle a, b, c \rangle \):

• Intersection of Planes: Solve the system of plane equations to find the line of intersection.

• Distance from Point to Line: where \( \vec{AP} \) is the vector from a point on the line to the given point.

• Distance from Point to Plane: for plane \( ax + by + cz + d = 0 \) and point \( (x_1, y_1, z_1) \).

• Example: Find the equation of the plane through (1,2,3) with normal vector \( \langle 4,5,6 \rangle \).

Section 6: Matching Concepts

This section likely includes conceptual questions matching definitions, properties, or examples to terms.

• Key Concepts: Vector operations, geometric interpretations, and properties of lines and planes.

• Example: Match the term "orthogonal projection" to its definition.

Chapter 13: Vector-Valued Functions and Motion in Space

Section 1: Vector Functions, Velocity, and Acceleration

Vector-valued functions describe curves in space, with derivatives representing velocity and acceleration.

• Vector Function: \( \vec{r}(t) = \langle x(t), y(t), z(t) \rangle \)

• Velocity: \( \vec{v}(t) = \vec{r}'(t) \)

• Acceleration: \( \vec{a}(t) = \vec{r}''(t) \)

• Constant Magnitude: If \( |\vec{r}(t)| \) is constant, then \( \vec{r}(t) \) and \( \vec{r}'(t) \) are orthogonal.

• Example: Given \( \vec{r}(t) = \langle \cos t, \sin t, 0 \rangle \), find \( \vec{v}(t) \) and \( \vec{a}(t) \).

Section 2: Projectile Motion and Initial Conditions

This section covers motion under constant acceleration, such as projectiles, and how to determine position functions from initial conditions.

• General Solution: where \( \vec{r}_0 \) is initial position, \( \vec{v}_0 \) is initial velocity, \( \vec{a} \) is constant acceleration.

• Example: Find \( \vec{r}(t) \) for a projectile launched from (0,0,0) with initial velocity \( \langle 2, 3, 4 \rangle \) and acceleration \( \langle 0, 0, -9.8 \rangle \).

Section 3: Arc Length and Curvature

Arc length measures the distance along a curve, and curvature quantifies how sharply a curve bends.

• Arc Length Formula:

• Curvature (\( \kappa \)):

• Arc Length as a Function: Sometimes, arc length \( s \) is used as a parameter for the curve.

• Example: Find the arc length of \( \vec{r}(t) = \langle t, t^2, t^3 \rangle \) from \( t=0 \) to \( t=1 \).

Section 4: Circle of Curvature, T, N, and Shortcuts

The circle of curvature at a point on a curve best approximates the curve near that point. The Tangent (T) and Normal (N) vectors describe the direction and change of the curve.

• Curvature (\( \kappa \)): As above, measures how quickly the curve changes direction.

• Tangent Vector (T):

• Normal Vector (N):

• Shortcut in 2D: If \( \vec{T} = \langle T_1, T_2 \rangle \), then \( \vec{N} = \langle -T_2, T_1 \rangle \).

• Example: For \( \vec{r}(t) = \langle t, t^2 \rangle \), find \( \vec{T} \) and \( \vec{N} \) at \( t=1 \).

Section 5: The Binormal Vector, Decomposition of Acceleration, and Curvature Formulas

This section introduces the binormal vector (B) and expresses acceleration in terms of tangent and normal components.

• Binormal Vector (B):

• Decomposition of Acceleration: where and

• Curvature Formulas: As above, and for plane curves:

• Example: For \( \vec{r}(t) = \langle \cos t, \sin t, t \rangle \), find \( \vec{T}, \vec{N}, \vec{B} \) at \( t=0 \).

Summary Table: Key Vector Operations and Formulas

Additional info: Some content and examples have been expanded for clarity and completeness based on standard Calculus III topics.