Vectors and Vector-Valued Functions in Calculus: Study Notes

12.1 - Vectors in the Plane

Definition and Representation of Vectors

Vectors are mathematical objects that have both magnitude (length) and direction. In the plane, a vector can be represented as an ordered pair or as a directed line segment from an initial point P to a terminal point Q.

• Component Form: If P = (a, b) and Q = (a_2, b_2), then the vector v from P to Q is v = (a_2 - a, b_2 - b).

• Magnitude: The length of vector v = (x, y) is given by: $\sqrt{x^2 + y^2}$

• Unit Vectors: Vectors of length 1, used to indicate direction. Common unit vectors in the plane are \( \mathbf{i} = (1, 0) \) and \( \mathbf{j} = (0, 1) \).

• Vector Addition: v + w = (v_1 + w_1, v_2 + w_2)

• Scalar Multiplication: c\mathbf{v} = (cv_1, cv_2)

• Triangle Inequality: $\|\mathbf{v} + \mathbf{w}\| \leq \|\mathbf{v}\| + \|\mathbf{w}\|$

Example:

If P = (2, -1) and Q = (5, 3), then v = (5-2, 3-(-1)) = (3, 4) and $\|\mathbf{v}\| = \sqrt{3^2 + 4^2} = 5$.

12.2 - 3D Space

Vectors in Three Dimensions

Vectors in 3D are represented as ordered triples and can describe positions and directions in space.

• Component Form: v = (a, b, c)

• Magnitude: $\|\mathbf{v}\| = \sqrt{a^2 + b^2 + c^2}$

• Standard Unit Vectors: \( \mathbf{i} = (1, 0, 0), \mathbf{j} = (0, 1, 0), \mathbf{k} = (0, 0, 1) \)

• Vector between Points: If P = (a, b, c) and Q = (a_2, b_2, c_2), then v = (a_2 - a, b_2 - b, c_2 - c)

Lines in Space

• Parametric Equations: A line through point P_0 = (x_0, y_0, z_0) in the direction of vector \mathbf{v} = (a, b, c) is given by: $x = x_0 + at$ $y = y_0 + bt$ $z = z_0 + ct$

• Checking if a Point Lies on a Line: Substitute the point into the parametric equations and solve for t. If a consistent t exists, the point lies on the line.

Planes in Space

• Equation of a Plane: A plane through point (x_0, y_0, z_0) with normal vector (a, b, c) is: $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$

12.3 - Dot Product and the Angle Between Two Vectors

Dot Product (Scalar Product)

The dot product of two vectors measures how much one vector extends in the direction of another.

• Definition: For \mathbf{a} = (a_1, a_2, a_3) and \mathbf{b} = (b_1, b_2, b_3): $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$

• Geometric Interpretation: $\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\|\|\mathbf{b}\| \cos \theta$ where $\theta$ is the angle between the vectors.

• Orthogonality: Vectors are orthogonal (perpendicular) if $\mathbf{a} \cdot \mathbf{b} = 0$.

• Projection: The projection of \mathbf{a} onto \mathbf{b} is: $\mathrm{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|^2} \right) \mathbf{b}$

Example:

Let \mathbf{a} = (1, 2, 3) and \mathbf{b} = (4, -5, 6). Then $\mathbf{a} \cdot \mathbf{b} = 1 \times 4 + 2 \times (-5) + 3 \times 6 = 4 - 10 + 18 = 12$

12.4 - Cross Product

Definition and Properties

The cross product of two vectors in 3D produces a vector orthogonal to both original vectors.

• Definition: For \mathbf{a} = (a_1, a_2, a_3) and \mathbf{b} = (b_1, b_2, b_3): $\mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)$

• Magnitude: $\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\|\|\mathbf{b}\| \sin \theta$

• Direction: Determined by the right-hand rule.

• Properties:

  • Anticommutative: $\mathbf{a} \times \mathbf{b} = - (\mathbf{b} \times \mathbf{a})$

  • Distributive: $\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}$

  • If vectors are parallel, $\mathbf{a} \times \mathbf{b} = \mathbf{0}$

Example:

Let \mathbf{a} = (1, 2, 3) and \mathbf{b} = (4, 5, 6). $\mathbf{a} \times \mathbf{b} = (2 \times 6 - 3 \times 5, 3 \times 4 - 1 \times 6, 1 \times 5 - 2 \times 4) = (12 - 15, 12 - 6, 5 - 8) = (-3, 6, -3)$

12.5 - Planes in 3-Space

Equation of a Plane

A plane in 3D is determined by a point and a normal vector.

• General Equation: $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$

• Normal Vector: The vector (a, b, c) is perpendicular to the plane.

• Point in Plane: (x_0, y_0, z_0) is a specific point on the plane.

• Intersection with Lines: Substitute the parametric equations of a line into the plane equation to find intersection points.

Example:

Find the equation of the plane through (2, 1, 5) with normal vector (1, -1, 2): $1(x - 2) - 1(y - 1) + 2(z - 5) = 0$

13.1 - Vector-Valued Functions

Definition and Parameterization

A vector-valued function assigns a vector to each value of a parameter, often representing a curve in space.

• General Form: $\mathbf{r}(t) = (x(t), y(t), z(t))$

• Parameterization: Describes a curve by expressing coordinates as functions of a parameter t.

• Examples:

  • Circle in the plane: $x(t) = r \cos t,\ y(t) = r \sin t$

  • Helix: $x(t) = a \cos t,\ y(t) = a \sin t,\ z(t) = bt$

Example:

Parameterize the circle of radius 5 centered at the origin: $\mathbf{r}(t) = (5 \cos t, 5 \sin t)$

13.2 - Calculus of Vector-Valued Functions

Differentiation and Integration

Vector-valued functions can be differentiated and integrated component-wise.

• Derivative: $\mathbf{r}'(t) = (x'(t), y'(t), z'(t))$

• Velocity: The derivative of position vector, $\mathbf{v}(t) = \mathbf{r}'(t)$

• Acceleration: The derivative of velocity, $\mathbf{a}(t) = \mathbf{r}''(t)$

• Integration: $\int \mathbf{r}(t)\,dt = (\int x(t)\,dt, \int y(t)\,dt, \int z(t)\,dt)$

• Product Rules:

  • For scalar function f(t) and vector function \( \mathbf{r}(t) \): $\frac{d}{dt}[f(t)\mathbf{r}(t)] = f'(t)\mathbf{r}(t) + f(t)\mathbf{r}'(t)$

  • For dot and cross products: $\frac{d}{dt}[\mathbf{r}(t) \cdot \mathbf{s}(t)] = \mathbf{r}'(t) \cdot \mathbf{s}(t) + \mathbf{r}(t) \cdot \mathbf{s}'(t)$ $\frac{d}{dt}[\mathbf{r}(t) \times \mathbf{s}(t)] = \mathbf{r}'(t) \times \mathbf{s}(t) + \mathbf{r}(t) \times \mathbf{s}'(t)$

Example:

If $\mathbf{r}(t) = (t, t^2, t^3)$, then $\mathbf{r}'(t) = (1, 2t, 3t^2)$

13.3 - Arc Length and Speed

Arc Length of a Curve

The arc length of a curve described by a vector-valued function $\mathbf{r}(t)$ from $t = a$ to $t = b$ is:

• $s = \int_a^b \|\mathbf{r}'(t)\|\,dt$

• Speed: The magnitude of the velocity vector, $\|\mathbf{r}'(t)\|$

Example:

For $\mathbf{r}(t) = (\cos t, \sin t)$, $\mathbf{r}'(t) = (-\sin t, \cos t)$, so $\|\mathbf{r}'(t)\| = 1$. The arc length from $t = 0$ to $t = 2\pi$ is $\int_0^{2\pi} 1\,dt = 2\pi$ (the circumference of the unit circle).

Summary Table: Key Vector Operations

Operation	Formula	Result
Vector Addition	$(a_1, a_2, a_3) + (b_1, b_2, b_3)$	$(a_1 + b_1, a_2 + b_2, a_3 + b_3)$
Scalar Multiplication	$c(a_1, a_2, a_3)$	$(ca_1, ca_2, ca_3)$
Dot Product	$\mathbf{a} \cdot \mathbf{b}$	Scalar
Cross Product	$\mathbf{a} \times \mathbf{b}$	Vector orthogonal to both
Magnitude	$\|\mathbf{a}\|$	Length (scalar)
Projection	$\mathrm{proj}_{\mathbf{b}} \mathbf{a}$	Vector in direction of $\mathbf{b}$

Additional info: Some context and notation were inferred and clarified for completeness, including standard forms for vector equations, properties, and calculus operations on vector-valued functions.