Vectors and Dot Product - Lecture Notes

Vectors in R^2 and Basic Operations

• Vector: A fundamental mathematical object characterized by both magnitude (length or size) and direction. It represents a displacement or a force without a fixed starting point (a "free vector").

  • In a 2D Cartesian coordinate system (R^2), a vector is commonly represented as an ordered pair $(x, y)$, denoting its components along the x and y axes.

  • In a 3D Cartesian system (R^3), it is represented as $(x, y, z)$.

• Displacement Vector: The vector from an initial point P to a terminal point Q, denoted as $\vec{PQ}$.

  • It is computed by subtracting the coordinates of the initial point from the coordinates of the terminal point: $\vec{PQ} = Q - P$. This operation determines the change in each coordinate required to move from P to Q.

  • Example: If $P = (-1, 3)$ and $Q = (3, 7)$, then the displacement vector is: $\vec{PQ} = Q - P = (3 - (-1),\ 7 - 3) = (4, 4)$.

• Convention: Vectors are typically written using angle-bracket notation, e.g., $\lt v1, v2 \gt$ in R^2 or $\lt v1, v2, v3 \gt$ in R^3, to distinguish them from points $(p1, p2)$. However, ordered pair notation is also commonly used interchangeably, especially when context makes it clear it's a vector.

Vector Representation and Components

• A vector $v$ in R^2 can be uniquely expressed by its components as $v = \lt v1, v2 \gt$.

• Similarly, a vector $v$ in R^3 is $v = \lt v1, v2, v3 \gt$.

• Component Form using Basis Vectors: Many vectors are also written as a linear combination of standard basis vectors, which are orthogonal (perpendicular) and unit length vectors aligned with the coordinate axes:

  • In R^2: The standard basis vectors are $i = \lt 1, 0 \gt$ (unit vector along the positive x-axis) and $j = \lt 0, 1 \gt$ (unit vector along the positive y-axis). Any vector $v = \lt v1, v2 \gt$ can then be written as $v = v1 \, i + v2 \, j$. This form emphasizes how much the vector extends in each dimension.

  • In R^3: The standard basis vectors are $i = \lt 1, 0, 0 \gt$, $j = \lt 0, 1, 0 \gt$, and $k = \lt 0, 0, 1 \gt$. A vector $v = \lt v1, v2, v3 \gt$ can be written as $v = v1 \, i + v2 \, j + v3 \, k$. This decomposition simplifies many vector calculations.

Vector Operations

• Vector Addition (Tip-to-Tail Rule / Parallelogram Rule):

  • For two vectors $v = \lt v1, v2 \gt$ and $w = \lt w1, w2 \gt$, their sum is found by adding their corresponding components: $v + w = \lt v1 + w1, v2 + w2 \gt$.

  • Geometric Interpretation:

    • Tip-to-Tail: If you place the tail of vector $w$ at the head (tip) of vector $v$, their sum $v + w$ is the vector from the tail of $v$ to the head of $w$.

    • Parallelogram Rule: If $v$ and $w$ share the same initial point, their sum $v + w$ is the diagonal of the parallelogram formed by $v$ and $w$. The other diagonal, $w - v$, represents a different vector.

  • Properties:

    • Addition is commutative: $v + w = w + v$. The order of addition does not affect the resultant vector.

    • Displacement vectors can be composed by adding displacements, representing consecutive movements.

• Scalar Multiplication:

  • For a scalar (a real number) $\alpha$ and a vector $v = \lt v1, v2 \gt$, scalar multiplication scales each component of the vector: \alpha v = \lt \alpha v1, \alpha v2 \gt \text{ (in R^2)} . Similarly for R^3: \alpha \lt v1, v2, v3 \gt = \lt \alpha v1, \alpha v2, \alpha v3 \gt \text{ (in R^3)} .

  • Geometric Interpretation:

    • The length of the new vector becomes $|\alpha|$ times the length of the original vector ($||\alpha v|| = |\alpha| ||v||$).

    • If \alpha > 0, the direction of the vector is preserved.

    • If \alpha < 0, the vector reverses its direction (points in the opposite way) while its length is scaled by $|\alpha|$.

Unit Vectors and Direction

• A unit vector is a vector with a magnitude (length) of 1. It is primarily used to specify direction without implying a specific magnitude.

• Any nonzero vector $w$ can be expressed as the product of its magnitude and a unit vector in its direction: $w = ||w|| u$, where $u$ is the unit vector (meaning $||u|| = 1$).

• In R^2, a unit vector can be conveniently parametrized by an angle $\theta$ with respect to the positive x-axis: $u = (\cos \theta, \sin \theta)$. This form provides a direct way to specify direction using polar coordinates.

• This effectively separates the concept of direction from magnitude, which is crucial in many physics and engineering applications (e.g., normalized vectors in graphics).

Parametric Curves and Kinematics (Vector-Valued Functions)

• A parametric curve describes the position of a point over time, typically represented by a vector-valued function. For example, in R^2, $P(t) = (x(t), y(t))$ or in R^3, $P(t) = (x(t), y(t), z(t))$. Here, $t$ is the parameter, often representing time.

• Velocity: The instantaneous velocity vector, $v(t)$, is the derivative of the position vector with respect to time: $v(t) = P'(t)$. It represents the instantaneous rate of change of position and is always tangent to the path of the curve at point $P(t)$. Its magnitude, $||v(t)||$, represents the instantaneous speed.

• Displacement over an interval: For a time interval $[t1, t2]$, the displacement, $\Delta P$, is the vector from the initial position to the final position: $\Delta P = P(t2) - P(t1)$.

• Average Velocity: Over an interval $[t1, t2]$, the average velocity is the total displacement divided by the time elapsed: $\bar{v} = \frac{\Delta P}{\Delta t} = \frac{P(t2) - P(t1)}{t2 - t1}$.

• Acceleration: The acceleration vector, $a(t)$, is the derivative of the velocity vector with respect to time: $a(t) = v'(t) = P''(t)$. It describes the rate of change of velocity, meaning it indicates how the speed, direction, or both are changing over time.

• Force and Mass (Newtonian Context): According to Newton's Second Law, the net force $\vec{F}$ acting on an object is equal to the product of its mass $m$ (a scalar) and its acceleration $\vec{a}$ (a vector): $\vec{F} = m \vec{a}$. This equation explicitly links vector quantities (Force, Acceleration) with a scalar (mass).

Magnitude (Length) of a Vector

• The magnitude or length of a vector is calculated using the Pythagorean theorem, representing the distance from the origin to the point $(v1, v2)$ (if the vector's tail is at the origin).

• For a vector $v = \lt v1, v2 \gt$ in R^2, its magnitude is given by: $||v|| = \sqrt{v1^2 + v2^2}$.

• For a vector $v = \lt v1, v2, v3 \gt$ in R^3, its magnitude is: $||v|| = \sqrt{v1^2 + v2^2 + v3^2}$.

• Example: For $v = (3, 4, 5)$, its magnitude is: $||v|| = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} = 5\sqrt{2}$.

• The magnitude is also known as the Euclidean norm of the vector, representing its length in Euclidean space.

Unit Vector and Direction of a Vector

• As noted previously, any nonzero vector $w$ can be uniquely expressed by its magnitude and a unit vector $u$ pointing in the same direction: $w = ||w|| u$. Here, $u$ captures the pure directional aspect of $w$.

• The unit vector $u$ in the direction of $w$ is obtained by normalizing $w$ (dividing it by its own magnitude): $u = \frac{w}{||w||}$, provided $w \neq 0$.

• In R^2, a unit vector can also be characterized by the angle $\theta$ it makes with the positive x-axis: $u = (\cos \theta, \sin \theta)$. This provides an alternative way to define the direction.

Dot Product (Inner Product) of Vectors

• The dot product (also known as the scalar product or inner product) is an algebraic operation that takes two vectors and returns a single scalar quantity.

• For vectors $a = \lt a1, a2, a3 \gt$ and $b = \lt b1, b2, b3 \gt$ in R^3, the dot product is calculated as the sum of the products of their corresponding components: $a \cdot b = a1 b1 + a2 b2 + a3 b3$.

• In R^2: For $a = \lt a1, a2 \gt$ and $b = \lt b1, b2 \gt$, the dot product is $a \cdot b = a1 b1 + a2 b2$. The R^3 definition simply adds the third component product.

• Algebraic Properties:

  • Commutative: $a \cdot b = b \cdot a$. The order of the vectors does not affect the result.

  • Distributive (Bilinear): $(a + b) \cdot c = a \cdot c + b \cdot c$. The dot product distributes over vector addition.

  • Scalar Multiplication Associativity: $\alpha (a \cdot b) = (\alpha a) \cdot b = a \cdot (\alpha b)$. A scalar factor can be moved freely within the dot product calculation.

  • Property with magnitude: $a \cdot a = ||a||^2$. The dot product of a vector with itself equals its magnitude squared.

• Relationship to the Angle Between Vectors (Geometric Interpretation):

  • A crucial property of the dot product is its connection to the angle between two non-zero vectors. If $a \neq 0$ and $b \neq 0$, then: $a \cdot b = ||a|| \, ||b|| \, \cos \phi$, where $\phi$ is the angle between vectors $a$ and $b$ ($0 \le \phi \le \pi$).

  • This formula allows us to calculate the angle between two vectors if their magnitudes and dot product are known, irrespective of their coordinate representation.

  • Example: Given $a = (2, 1, -3)$ and $b = (-2, 1, 1)$:

    • Calculate the dot product: $a \cdot b = 2(-2) + 1(1) + (-3)(1) = -4 + 1 - 3 = -6$.

    • Calculate magnitudes: $||a|| = \sqrt{2^2 + 1^2 + (-3)^2} = \sqrt{4 + 1 + 9} = \sqrt{14}$. $||b|| = \sqrt{(-2)^2 + 1^2 + 1^2} = \sqrt{4 + 1 + 1} = \sqrt{6}$.

    • Find the angle: $\cos \phi = \frac{a \cdot b}{||a|| \, ||b||} = \frac{-6}{\sqrt{14}\sqrt{6}} = \frac{-6}{\sqrt{84}} = -\frac{6}{2\sqrt{21}} = -\frac{3}{\sqrt{21}} \approx -0.6547$.

    • Thus, $\phi \approx \cos^{-1}(-0.6547) \approx 131^{\circ}$. This is an obtuse angle.

• Special Cases and Interpretations:

  • Perpendicular (Orthogonal) Vectors: If $a$ and $b$ are perpendicular, the angle between them is $90^{\circ}$ ($\pi/2$ radians), so $\cos \phi = 0$. Therefore, $a \cdot b = 0$. This is a powerful test for orthogonality.

  • Acute Angle: If a \cdot b > 0, then \cos \phi > 0, implying $\phi$ is an acute angle (0 \le \phi < 90^{\circ}).

  • Obtuse Angle: If a \cdot b < 0, then \cos \phi < 0, implying $\phi$ is an obtuse angle (90^{\circ} < \phi \le 180^{\circ}).

  • Zero Vector: If either vector is the zero vector, the dot product is 0, and the angle between them is undefined.

  • Vector Projection: The dot product is also used to calculate the scalar projection of one vector onto another, which represents how much one vector extends in the direction of another. The projection of $b$ onto $a$ is given by $\text{proj}_a b = \left( \frac{a \cdot b}{||a||^2} \right) a$.

Law of Cosines (Vector Form)

• A fundamental identity linking vector magnitudes and the dot product is the vector form of the Law of Cosines:
  $||a - b||^2 = ||a||^2 + ||b||^2 - 2\,a \cdot b$.

• This formula relates the square of the magnitude of the difference between two vectors to the squares of their individual magnitudes and their dot product.

• It mathematically describes the third side of a triangle formed by vectors $a$, $b$, and $a - b$. By rearranging this equation, one can derive the angle formula: $\cos \phi = \frac{||a||^2 + ||b||^2 - ||a - b||^2}{2||a|| \, ||b||}$. Alternatively, substituting $a \cdot b = ||a|| ||b|| \cos \phi$ directly yields the familiar Law of Cosines.

• Consequences:

  • If you know the magnitudes $||a||$ and $||b||$, and the dot product $a \cdot b$, you can directly compute the angle $\phi$ between them without needing their explicit coordinates.

  • This identity provides a robust way to analyze geometric configurations using vector algebra.

Quick Computation Checks and Examples

• Example Recap for Dot Product and Magnitude:

  • Given $a = (2, 1, -3)$ and $b = (-2, 1, 1)$:

    • Dot Product: $a \cdot b = -6$

    • Magnitudes: $||a|| = \sqrt{14}, ||b|| = \sqrt{6}$

    • Cosine of angle: $\cos \phi = -6 / (\sqrt{14} \sqrt{6}) = -3 / \sqrt{21} \approx -0.6547$

    • Angle: $\phi \approx 131^{\circ}$ (indicating an obtuse angle, consistent with a negative dot product).

• Example for Magnitude in R^2:

  • For $v = (3, 4)$: $||v|| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

• Example for Displacement and Velocity in Kinematics:

  • If $P(t)$ is a position vector-valued function, the instantaneous velocity $v(t) = P'(t)$.

  • Over an interval $[t1, t2]$, the displacement is $\Delta P = P(t2) - P(t1)$. The average velocity is $\Delta P / (t2 - t1)$, which is a vector pointing in the direction of the net displacement.

Summary of Key Formulas

• Displacement: $\vec{PQ} = Q - P$

• Magnitude: For $v = \lt v1, v2 \gt$ or $v = \lt v1, v2, v3 \gt$, $||v|| = \sqrt{v1^2 + v2^2 (+ v3^2)}$

• Unit Vector (Normalization): $u = \frac{v}{||v||} \quad \text{for } v \neq 0$; conversely, $v = ||v|| u$

• Dot Product: For $a = \lt a1, a2 \gt$ and $b = \lt b1, b2 \gt$ (or with $a3, b3$ components), $a \cdot b = a1 b1 + a2 b2 (+ a3 b3)$

• Angle Between Vectors: For non-zero vectors $a$ and $b$, $\cos \phi = \frac{a \cdot b}{||a|| \, ||b||}$

• Law of Cosines (Vector Form): $||a - b||^2 = ||a||^2 + ||b||^2 - 2\, a \cdot b$

• Unit Vector in R^2 (Parametric): $u = (\cos \theta, \sin \theta)$

• Ray of a Unit Direction: If $a \neq 0$, then $a$ can be written as $a = ||a|| u$, where $u$ is the unit vector pointing in the direction of $a$.

Connections and Relevance

• These concepts are foundational to analytic geometry, providing a powerful way to represent and manipulate geometric objects (points, lines, planes) using algebraic methods (vectors).

• The dot product is particularly significant as it acts as a bridge between algebra and geometry, enabling detailed calculations related to angles, perpendicularity, and projections (e.g., finding the component of a force in a certain direction, determining work done by a force).

• Magnitude, unit vectors, and direction are critical for normalization procedures in various fields, such as computer graphics (e.g., calculating light reflection and surface normals), physics (e.g., specifying force directions), and engineering (e.g., material stress analysis).

• Kinematic quantities (displacement, velocity, acceleration) are inherently vector quantities, making this framework indispensable for analyzing motion in physics and mechanical engineering. They allow for a comprehensive understanding of how objects move and interact in space.

Ethical, Philosophical, and Practical Notes

• The use of Cartesian coordinate representation is a conventional choice that greatly simplifies vector arithmetic and linear algebra. While other coordinate systems (e.g., cylindrical, spherical) exist, the core vector operations remain analogous.

• A deep understanding of vectors fosters both intuitive qualitative thinking about physical phenomena (e.g., the direction of a force, the path of a projectile) and rigorous quantitative analysis in fields like fluid dynamics, electromagnetism, and structural analysis.

• When interpreting dot product results and the angles they imply, remember that a result of zero signifies perpendicularity, a positive result implies an acute angle, and a negative result indicates an obtuse angle. These geometric interpretations are vital for problem-solving and accurate model interpretation in scientific and engineering contexts, guiding decisions and predictions. Vectors thus provide a language for describing spatial relationships and dynamic processes with precision.