Vectors in Two Dimensions
Understand magnitude, direction, unit vectors, and position vectors in the two-dimensional plane.
What Is a Vector?
A vector is a quantity that has both magnitude (size) and direction. Vectors are used to represent quantities like displacement, velocity, and force. In contrast, a scalar has only magnitude (e.g., speed, temperature).
In two dimensions, we write vectors using column notation or component form. A vector a with horizontal component x and vertical component y is written as:
a = (x, y) = xi + yj
where i is the unit vector in the x-direction and j is the unit vector in the y-direction.
Magnitude and Direction
The magnitude (or length) of a vector a = (x, y) is found using Pythagoras' theorem:
|a| = √(x2 + y2)
The direction of a vector is the angle θ it makes with the positive x-axis, calculated using:
θ = tan−1(y / x)
Always check which quadrant the vector lies in to adjust the angle if needed.
Unit Vectors and Position Vectors
A unit vector has a magnitude of exactly 1. To find the unit vector in the direction of a, divide by its magnitude:
â = a / |a|
A position vector is the vector from the origin O to a point P. If P has coordinates (3, 4), the position vector of P is:
OP = 3i + 4j
Key Vocabulary
Magnitude
The length or size of a vector, calculated as √(x2 + y2).
Unit Vector
A vector with magnitude 1, found by dividing a vector by its magnitude.
Position Vector
A vector from the origin to a specific point in the plane.
Direction Angle
The angle a vector makes with the positive x-axis, found using inverse tangent.
Worked Examples
Find the magnitude of vector v = (3, 4).
Step 1: Apply the magnitude formula: |v| = √(x2 + y2).
Step 2: |v| = √(32 + 42) = √(9 + 16) = √25.
Answer: |v| = 5 units.
Find the unit vector in the direction of u = (5, 12).
Step 1: Find the magnitude: |u| = √(25 + 144) = √169 = 13.
Step 2: Divide each component by 13: û = (5/13, 12/13).
Answer: û = (5/13)i + (12/13)j.
Find the direction angle of vector w = (1, √3).
Step 1: θ = tan−1(√3/1) = tan−1(√3).
Step 2: Since both components are positive (first quadrant), θ = 60°.
Answer: The direction angle is 60° (or π/3 radians).
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the magnitude of the vector (6, 8)?
Question 2
What is the unit vector in the direction of (0, 5)?
Question 3
A vector has magnitude 13 and components (5, y). What is y?
Question 4
The position vector of point A(3, −2) is:
Question 5
What is the direction angle of the vector (1, 1)?
Key Concepts Summary
- ● A vector has both magnitude and direction, unlike a scalar.
- ● The magnitude of (x, y) is √(x2 + y2).
- ● A unit vector has magnitude 1 and is found by dividing by the magnitude.
- ● The direction angle is found using θ = tan−1(y/x), adjusted for the correct quadrant.
- ● A position vector runs from the origin to a given point.