Basic Concepts
Vector
A quantity that has both magnitude (size) and direction. Unlike scalars (which have only size), vectors need both pieces of information to be completely defined.
Example:
"Drive 5 km North" is a vector (magnitude = 5 km, direction = North). Just saying "5 km" is a scalar because it doesn't tell you which way to go.
Scalar
A quantity that has only magnitude (size), with no direction. Scalars are ordinary numbers that follow basic arithmetic rules.
Examples:
Distance (5 km), temperature (20°C), mass (2 kg), and time (3 seconds) are all scalars. They are just numbers without any sense of direction.
Magnitude
The length or size of a vector. For a vector starting at the origin and ending at point (x, y), the magnitude is calculated using the Pythagorean Theorem: |v| = √(x² + y²)
Example:
For vector v = 3i + 4j, the magnitude is |v| = √(3² + 4²) = √(9 + 16) = √25 = 5. This is also written as ‖v‖.
Direction
The path or bearing that a vector points towards. In 2D, direction is often described using compass directions (North, East, South, West) or an angle from the positive x-axis.
Example:
A vector pointing from the origin to (3, 3) has a direction of 45° from the positive x-axis (Northeast). The direction angle θ is found using: θ = atan2(y, x).
Origin
The starting point of a vector, usually labeled as O or (0, 0) in a coordinate system. All position vectors begin at the origin.
Example:
The point (0, 0) on a coordinate plane is the origin. A vector drawn from (0, 0) to (2, 3) is a position vector with magnitude √13.
Position Vector
A vector that starts at the origin (0, 0) and ends at a specific point. Its components directly represent the coordinates of that point.
Example:
The position vector to point P(5, 2) is OP = 5i + 2j. This vector starts at O(0, 0) and ends at P(5, 2).
Displacement Vector
A vector that represents the change in position from one point to another, calculated as: PQ = Q - P
Example:
If you move from point P(1, 2) to point Q(4, 5), the displacement vector is PQ = (4-1)i + (5-2)j = 3i + 3j. Magnitude = √18 ≈ 4.24 units.
Unit Vector
A vector with magnitude exactly equal to 1. Unit vectors are used to show direction without size. Any vector can be scaled to a unit vector: u = v / |v|
Example:
Vector v = 3i + 4j has magnitude |v| = 5. The unit vector is u = (3/5)i + (4/5)j = 0.6i + 0.8j, with magnitude 1.
Equal Vectors
Two vectors are equal if they have the same magnitude and direction, regardless of where they are positioned. This means: A = B if A_x = B_x AND A_y = B_y
Example:
A vector from (0, 0) to (2, 3) equals a vector from (5, 7) to (7, 10), because both have components 2i + 3j. They point the same direction and have the same length.
Negative Vector
The opposite of a vector, with the same magnitude but opposite direction. If v = ai + bj, then -v = -ai - bj
Example:
If v = 3i + 4j, then -v = -3i - 4j. Both have magnitude 5, but they point in opposite directions. v + (-v) = 0 (zero vector).
Zero Vector
A special vector with zero magnitude and no defined direction. Written as 0 or 0i + 0j. It represents no displacement or movement.
Example:
If a car drives 5 km East then 5 km West, the total displacement is the zero vector (you're back where you started). Also: v + (-v) = 0 for any vector v.
Parallel Vectors
Two vectors are parallel if one is a scalar multiple of the other. They point in the same direction (or opposite if the scalar is negative) but may have different lengths.
Example:
Vectors a = 2i + 4j and b = 6i + 12j are parallel because b = 3a (b is 3 times a). They point in the same direction but b is longer.
Vector Components
Component Form
A way of writing a vector using its horizontal (i) and vertical (j) parts. Written as: v = xi + yj or v = (x, y)
Example:
Vector v pointing to (3, 4) is written as v = 3i + 4j. The component 3i is horizontal, and 4j is vertical. This form makes calculations much easier.
i-component (x-component)
The horizontal part of a vector. It shows how far the vector moves left or right. Positive means moving right (East), negative means moving left (West).
Example:
In vector v = 3i + 4j, the i-component is 3. This means the vector moves 3 units to the right. For w = -2i + 5j, the i-component is -2, moving 2 units to the left.
j-component (y-component)
The vertical part of a vector. It shows how far the vector moves up or down. Positive means moving up (North), negative means moving down (South).
Example:
In vector v = 3i + 4j, the j-component is 4. This means the vector moves 4 units upward. For w = 2i - 3j, the j-component is -3, moving 3 units downward.
Component Decomposition
Breaking a vector into its horizontal and vertical parts. The original vector forms the hypotenuse of a right triangle, with components as the two sides.
Example:
Vector v = 3i + 4j is shown as a right triangle: horizontal side = 3, vertical side = 4, hypotenuse (the vector) has magnitude √(3² + 4²) = 5.
Column Vector Notation
A way of writing vectors as a column of numbers, where the top number is the x-component and the bottom is the y-component. Written as: [x] or (x, y) or sometimes ⎧x⎫
⎩y⎭
⎩y⎭
Example:
Vector v = 3i + 4j can be written in column form as ⎧3⎫⎩4⎭ or sometimes (3, 4) in coordinate notation.
Magnitude from Components
Using the Pythagorean Theorem to find the length of a vector from its components: |v| = √(x² + y²)
Example:
For v = 5i + 12j: |v| = √(5² + 12²) = √(25 + 144) = √169 = 13. The components form a 5-12-13 right triangle.
Pythagorean Theorem
For any right triangle with sides a and b and hypotenuse c: a² + b² = c². Vector components always form a right triangle with the vector as the hypotenuse.
Example:
A 3-4-5 triangle: 3² + 4² = 9 + 16 = 25 = 5². Vector v = 3i + 4j has magnitude 5, and its components are the perpendicular sides.
Direction Angle
The angle measured counterclockwise from the positive x-axis to the vector, usually in the range 0° to 360°. Calculated using: θ = atan2(y, x)
Example:
For v = 3i + 3j: θ = atan2(3, 3) ≈ 45°. For w = -3i + 3j in Quadrant II: θ ≈ 135°. Always check the quadrant!
Quadrant
One of the four regions of the coordinate plane created by the x and y axes. Quadrant I (both positive), II (x negative, y positive), III (both negative), IV (x positive, y negative).
Example:
Vector 3i + 4j is in Quadrant I (45° angle, both components positive). Vector -3i + 4j is in Quadrant II (135° angle, pointing upper left).
Vector Operations
Vector Addition
Combining two vectors to create a resultant vector. Add component by component: (a₁i + a₂j) + (b₁i + b₂j) = (a₁+b₁)i + (a₂+b₂)j
Example:
(2i + 3j) + (1i + 4j) = (2+1)i + (3+4)j = 3i + 7j. Geometrically: place the tail of the second vector at the head of the first; the result goes from the starting point to the final point.
Vector Subtraction
Subtracting one vector from another. Rewrite as adding the negative: A - B = A + (-B). Then add components: (a₁-b₁)i + (a₂-b₂)j
Example:
(5i + 7j) - (2i + 3j) = (5-2)i + (7-3)j = 3i + 4j. It's like reversing the second vector and then adding it to the first.
Scalar Multiplication
Multiplying a vector by a number (scalar) to change its length or direction: k(ai + bj) = (ka)i + (kb)j
Example:
2(3i + 4j) = 6i + 8j. This doubles the vector's length. If k is negative, the vector reverses direction. 0.5(3i + 4j) = 1.5i + 2j (halves the length).
Linear Combination
A sum of scalar multiples of vectors. Given vectors a and b: pa + qb where p and q are scalars. Any vector in the plane can be expressed this way.
Example:
If a = 2i + j and b = i + 3j, then 2a - 3b = 2(2i + j) - 3(i + 3j) = 4i + 2j - 3i - 9j = i - 7j.
Resultant Vector
The final vector obtained by adding or combining other vectors. It represents the net effect of all the individual vectors acting together.
Example:
If you walk 3 km East then 4 km North, your resultant displacement is 5 km Northeast (vector 3i + 4j with magnitude 5 km).
Triangle Law
A geometric method for adding vectors: place the tail of the second vector at the head (tip) of the first; the resultant goes from the starting point to the final point, forming a triangle.
Example:
Add vectors A and B using the triangle law: draw A, then from its endpoint, draw B. The result R goes from the start of A to the end of B, and R = A + B.
Parallelogram Law
An alternative geometric method for adding vectors: place both vectors at the same starting point and complete a parallelogram; the diagonal from the starting point to the opposite corner is the resultant.
Example:
For vectors A and B starting at the origin, complete the parallelogram. The diagonal equals A + B. Useful for understanding forces acting at the same point.
Dot Product (not covered in basic course)
A way to multiply two vectors to get a scalar (single number): a · b = a₁b₁ + a₂b₂ = |a||b|cos(θ). Useful for finding angles between vectors.
Example:
For a = 2i + 3j and b = 1i + 1j: a · b = (2)(1) + (3)(1) = 5. If a · b = 0, the vectors are perpendicular.
Real-World Applications
Displacement vs Distance
Distance is the total path length (scalar, always positive). Displacement is the straight-line change in position (vector, can be positive or negative).
Example:
Drive 5 km East then 3 km West: distance = 5 + 3 = 8 km. Displacement = (5-3)i = 2i km (2 km East). Distance tells how far you traveled; displacement tells where you ended up.
Velocity
A vector quantity showing how fast something is moving and in which direction. Velocity = Displacement / Time. Unlike speed (scalar), velocity includes direction.
Example:
An airplane flies 500 km North in 1 hour. Its velocity is 500 km/h North. If we say speed is 500 km/h, we lose the direction information.
Force
A vector quantity that pushes or pulls on an object, with both magnitude (newtons) and direction. Multiple forces can be combined using vector addition.
Example:
A 10 N force East and a 5 N force North combine to give a resultant force of √(10² + 5²) = √125 ≈ 11.2 N at about 27° North of East.
Equilibrium
A state where all forces are balanced, so the resultant force is zero. An object in equilibrium has no net force acting on it and won't accelerate.
Example:
A picture hanging on a wall is in equilibrium: tension pulls up, gravity pulls down, and they're equal. The resultant is the zero vector.
Wind Correction (Airplane Navigation)
When an airplane must adjust its heading to account for wind. The plane's velocity vector and wind vector combine to give the resultant velocity (actual direction traveled).
Example:
Plane heading: 200 km/h East. Wind: 50 km/h North. Resultant velocity = √(200² + 50²) ≈ 206 km/h at about 14° North of East (actual path).
Object on an Incline
Gravity force can be decomposed into components parallel and perpendicular to the incline. The parallel component causes the object to slide; the perpendicular component presses into the surface.
Example:
A box on a 30° incline has weight 100 N downward. Component along incline ≈ 50 N (pulls it down). Component perpendicular ≈ 87 N (presses into surface).
Navigation & Bearing
Vectors are used to describe positions, headings, and distances in navigation. Bearing is the direction angle, often measured clockwise from North (unlike math's counterclockwise from East).
Example:
"100 km at bearing 45°" means 100 km from North towards East. In math coordinates, this is the vector pointing at angle 45° from the East direction.
Mathematical Notation
Vector Notation Symbols
Different ways to write vectors in mathematics. Common notations include: boldface v, arrow →v, hats v̂, and bold letters a.
Example:
All of these mean the same thing: v, →v, (2, 3), 2i + 3j, and ⎧2⎫⎩3⎭
Magnitude Notation
The magnitude of vector v is written as: |v| or ‖v‖ or sometimes just the letter without the vector mark. Both mean the "length" or "size" of the vector.
Example:
If v = 3i + 4j, then |v| = 5 and ‖v‖ = 5. Both notations mean the same thing: the magnitude is 5 units.
i and j (Standard Basis Vectors)
Unit vectors: i points East (1, 0) and j points North (0, 1). Any vector can be written as v = xi + yj where x and y are scalars. These form the "basis" for 2D vectors.
Example:
Vector v = 3i + 4j means "move 3 units in the i-direction and 4 units in the j-direction." i and j are the building blocks of all 2D vectors.
Coordinate Notation
Writing vectors as ordered pairs: (x, y) or sometimes with brackets [x, y]. The first number is the x-component, the second is the y-component.
Example:
Vector v = 3i + 4j can be written as (3, 4) in coordinate notation. Be careful not to confuse this with a point: the vector goes from (0,0) to (3,4).
Greek Letters in Vectors
Greek letters are commonly used for angles and parameters: θ (theta) for angle, λ (lambda) and μ (mu) for scalar parameters, and Σ (sigma) for summation.
Example:
"Find θ such that vector v makes angle θ with the x-axis." Or "Find λ such that λa + μb = c." These Greek letters are just placeholders for unknown numbers.
Subscript Notation
Using subscripts to denote components: v₁ for the first component (x-component) and v₂ for the second component (y-component). So v = v₁i + v₂j.
Example:
For vector a = 5i + 2j, we write a₁ = 5 and a₂ = 2. This notation is especially useful in formulas: |a| = √(a₁² + a₂²).
atan2 Function
A function that finds the angle of a vector from its components: θ = atan2(y, x). Unlike arctan, it correctly handles all four quadrants.
Example:
For v = 3i + 3j: θ = atan2(3, 3) = 45°. For w = -3i + 3j: θ = atan2(3, -3) = 135° (Quadrant II). atan2 always gives the right angle!
Matrix Notation
Vectors can be written as matrices (single column or row): v = ⎧3⎫
⎩4⎭ (column matrix) or v = [3 4] (row matrix). Useful for calculations and computer programming.
⎩4⎭ (column matrix) or v = [3 4] (row matrix). Useful for calculations and computer programming.
Example:
Vector v = 3i + 4j as column matrix: ⎧3⎫⎩4⎭. Matrix operations are used in computer graphics and linear algebra.