Fundamental Properties of Vectors: A Mini-Textbook Study Guide

Fundamental Properties of Vectors

Equality of Two Vectors

Vectors are mathematical objects characterized by both magnitude and direction. Two vectors \( \vec{A} \) and \( \vec{B} \) are considered equal if they have the same magnitude and point in the same direction, regardless of their initial position.

• Definition: \( \vec{A} = \vec{B} \) if and only if their magnitudes and directions are identical.

• Example: Two arrows of equal length pointing north are equal vectors.

Adding Vectors

Vector addition combines two or more vectors to produce a resultant vector. There are two main graphical methods for vector addition:

• Graphical (Head-to-Tail) Method: Place the tail of the second vector at the head of the first; the resultant vector is drawn from the tail of the first to the head of the last.

• Parallelogram Method: Place both vectors tail-to-tail, construct a parallelogram, and the diagonal represents the resultant vector.

• Commutativity: The order of addition does not affect the resultant: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$

• Example: Adding displacement vectors for a car traveling north and then west.

Negative of a Vector and Subtracting Vectors

The negative of a vector reverses its direction but keeps its magnitude. Subtracting a vector is equivalent to adding its negative.

• Definition: \( -\vec{A} \) is a vector with the same magnitude as \( \vec{A} \) but opposite direction.

• Subtraction: \( \vec{A} - \vec{B} = \vec{A} + (-\vec{B}) \)

• Example: If \( \vec{B} \) points east, \( -\vec{B} \) points west.

Multiplying a Vector by a Scalar

Multiplying a vector by a scalar changes its magnitude but not its direction (unless the scalar is negative).

• Positive Scalar: \( m\vec{A} \) has magnitude \( m|\vec{A}| \) and the same direction as \( \vec{A} \).

• Negative Scalar: \( -m\vec{A} \) has magnitude \( m|\vec{A}| \) and the opposite direction.

• Example: Doubling a velocity vector doubles the speed in the same direction.

Vector Components and Unit Vectors

Components of a Vector

Any vector in a plane can be decomposed into components along the x and y axes.

• Component Form: \( \vec{A} = A_x \hat{i} + A_y \hat{j} \)

• Formulas: $A_x = A \cos \theta$ $A_y = A \sin \theta$ $A = \sqrt{A_x^2 + A_y^2}$ $\tan \theta = \frac{A_y}{A_x}$ $\theta = \arctan \left( \frac{A_y}{A_x} \right)$

• Example: A force of 10 N at 30° above the x-axis has components $A_x = 10 \cos 30°$, $A_y = 10 \sin 30°$.

Unit Vectors

Unit vectors are dimensionless vectors with magnitude 1, used to specify direction along coordinate axes.

• Notation: \( \hat{i} \) (x-axis), \( \hat{j} \) (y-axis), \( \hat{k} \) (z-axis)

• Properties: $|\hat{i}| = |\hat{j}| = |\hat{k}| = 1$

• Example: \( \vec{A} = 3\hat{i} + 4\hat{j} \) is a vector in the xy-plane.

Vectors in Three Dimensions

Vectors can be extended to three dimensions using the z-axis.

• General Form: \( \vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} \)

• Resultant Vector: \( \vec{R} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k} \)

• Magnitude: $R = \sqrt{R_x^2 + R_y^2 + R_z^2}$

Products of Two Vectors

Scalar (Dot) Product

The scalar product (dot product) of two vectors yields a scalar quantity and is defined as:

• Formula: $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta$

• Properties:

  • Commutative: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$

  • If vectors are perpendicular: $\theta = 90°$, $\vec{A} \cdot \vec{B} = 0$

  • If vectors are parallel: $\theta = 0°$, $\vec{A} \cdot \vec{B} = AB$

  • If vectors are anti-parallel: $\theta = 180°$, $\vec{A} \cdot \vec{B} = -AB$

• Component Form: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$

• Example: Work done by a force: $W = \vec{F} \cdot \vec{d}$

Vector (Cross) Product

The vector product (cross product) of two vectors yields a new vector perpendicular to both original vectors.

• Formula: $\vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin \theta \, \hat{n}$, where $\hat{n}$ is a unit vector perpendicular to the plane of \( \vec{A} \) and \( \vec{B} \).

• Properties:

  • Anti-commutative: $\vec{A} \times \vec{B} = - (\vec{B} \times \vec{A})$

  • If vectors are parallel or anti-parallel: $\sin 0° = 0$, $\vec{A} \times \vec{B} = 0$

• Determinant Method: $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$

• Example: Torque: $\vec{\tau} = \vec{r} \times \vec{F}$

Unit Vector Products

Unit vectors follow specific multiplication rules:

• Dot Product: $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$; $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$

• Cross Product:

  • $\hat{i} \times \hat{j} = \hat{k}$

  • $\hat{j} \times \hat{k} = \hat{i}$

  • $\hat{k} \times \hat{i} = \hat{j}$

  • Reverse order changes sign: $\hat{j} \times \hat{i} = -\hat{k}$, etc.

Worked Example: Displacement Calculation

Suppose a car travels 20 km north and then 35 km at 60° west of north. To find the resultant displacement:

• Given: $|\vec{A}| = 20$ km, $|\vec{B}| = 35$ km, $\alpha = 120°$

• Law of Cosines: $R^2 = A^2 + B^2 - 2AB \cos \alpha$ $R^2 = 20^2 + 35^2 - 2 \times 20 \times 35 \cos 120°$ $R = 48.2$ km

• Law of Sines (for direction): $\frac{\sin \beta}{|\vec{B}|} = \frac{\sin 120°}{48.2}$ $\beta = 38.9°$ west of north

Summary Table: Vector Operations

Operation	Formula	Result Type	Key Properties
Addition	$\vec{A} + \vec{B}$	Vector	Commutative, associative
Subtraction	$\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$	Vector	Reverse direction for negative
Scalar (Dot) Product	$\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta$	Scalar	Commutative, zero if perpendicular
Vector (Cross) Product	$\vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin \theta \, \hat{n}$	Vector	Anti-commutative, zero if parallel
Unit Vector	$|\hat{i}| = |\hat{j}| = |\hat{k}| = 1$	Vector	Specifies direction only

Additional info: These notes cover fundamental vector operations and properties, which are foundational in physics and mathematics, but not directly related to microeconomics. However, understanding vectors is essential for quantitative analysis in economics, especially in optimization and graphical analysis.