In mathematics and physics, a vector is a fundamental object that is defined by two key attributes: magnitude and direction. Imagine a straight line segment in space; a vector is essentially this segment with a specified direction. The length of the segment represents the vector’s magnitude, while an arrow at one end indicates its direction. Think of it as a directed line segment, extending from its tail to its head.
Fundamentally, two vectors are considered identical if and only if they possess the same magnitude and direction. This crucial property implies that if you were to move a vector to a different location without rotating it, the vector remains unchanged. Its position in space is irrelevant; only its magnitude and direction define it.
Vectors are ubiquitous in science and engineering. Consider force and velocity as prime examples. Both inherently possess a direction and a strength or speed. In the case of force, the magnitude indicates the force’s strength, while for velocity, it signifies the speed.
We use specific notations to represent vectors, typically employing boldface letters such as a or b. In handwriting, where bolding is less convenient, an arrow above the letter (like $vec{a}$ or $vec{b}$) is often used, or other distinguishing marks. Here, we will consistently use boldface. The magnitude of a vector a is denoted by $||{bf{a}}||$. To differentiate a number from a vector, we use the term scalar for a simple numerical value. Scalars are represented in italics, for instance, $a$ or $b$.
Explore the interactive tool below to grasp the magnitude and direction of a vector dynamically. Observe that repositioning the vector doesn’t alter it, as its magnitude and direction remain constant. However, manipulating its head or tail to stretch or rotate it will indeed change its magnitude or direction. (This tool also displays vector coordinates, which are discussed in detail on another page.)
Visualizing Vector Magnitude and Direction. The blue arrow here represents a vector a. Its magnitude and direction, the defining characteristics of a vector, are visually represented by a red bar and a green arrow, respectively. The red bar’s length corresponds to the magnitude $||{bf{a}}||$ of vector a. The green arrow, always of unit length, points in the direction of vector a. An exception arises with the zero vector (the only vector with zero magnitude), where direction is undefined. Experiment by dragging either end of a to modify it. Dragging the middle of a simply repositions it without altering its fundamental properties.
Learn more about this interactive tool.
There’s a noteworthy exception to the rule that vectors have direction: the zero vector. Represented as 0, it’s the unique vector with zero magnitude. Having no length, it inherently lacks a defined direction. Since only one vector possesses zero length, we refer to it as the zero vector. Learn more about the zero vector.
Vector Operations
We can perform operations on vectors using geometric methods, independent of any coordinate system. Here, we will define fundamental operations including addition, subtraction, and scalar multiplication. For vector multiplication, we explore two distinct types on separate pages: the dot product and the cross product.
Addition of Vectors {#addition-of-vectors}
To add two vectors, a and b, we perform vector addition, denoted as a + b, geometrically. Imagine moving vector b so that its tail coincides with the head of vector a. This translation doesn’t change vector b itself. Then, the vector sum a + b is the vector drawn from the tail of a to the head of b.
See a visual explanation of vector addition.
Vector addition is fundamental to understanding how forces and velocities combine. For instance, consider a car traveling north at 20 mph. If a passenger throws an object eastward at 20 mph relative to the car, the object’s velocity relative to the ground is a vector sum. The two velocity vectors (northward car velocity and eastward object velocity) form two sides of a right triangle. The resultant velocity, the hypotenuse, is in a northeast direction. The magnitude of this resultant velocity, or the object’s speed relative to the ground, is $sqrt{20^2+20^2}=20sqrt{2}$ mph.
Vector addition adheres to two important laws:
- Commutative Law: The order of addition doesn’t affect the result: ${bf{a}} + {bf{b}} = {bf{b}} + {bf{a}}$. This is also known as the parallelogram law. Imagine a parallelogram where vectors a and b form adjacent sides. Then, both a + b and b + a represent the same diagonal of the parallelogram.
- Associative Law: When adding three vectors, the grouping doesn’t matter: $({bf{a}} + {bf{b}}) + {bf{c}} = {bf{a}} + ({bf{b}} + {bf{c}})$.
Explore the applet below to visualize these properties of vector addition. (This applet also shows vector coordinates, discussed further in another resource.)
Interactive Vector Addition. The sum a + b of vector a (blue arrow) and vector b (red arrow) is shown as the green arrow. Since vectors are position-independent, both blue arrows represent the same vector a, and both red arrows represent b. The sum a + b can be constructed by placing the tail of b at the head of a, or vice versa. Together, these constructions form a parallelogram, with a + b as a diagonal. This illustrates the commutative law ${bf{a}} + {bf{b}} = {bf{b}} + {bf{a}}$, often called the parallelogram law. Drag the yellow points to change a and b and observe the sum.
Access the interactive vector addition applet.
Vector Subtraction {#vector-subtraction}
Before defining subtraction, let’s define the opposite vector, –a. The vector –a has the same magnitude as a but points in the exactly opposite direction.
Visual representation of opposite vectors.
Vector subtraction, b – a, is defined as adding b to the opposite of a: ${bf{b}} – {bf{a}} = {bf{b}} + (-{bf{a}})$. Geometrically, this involves reversing the direction of a and then applying the vector addition rules. Consider vector x in the figure; it represents b – a. Notice that this is equivalent to saying ${bf{a}} + {bf{x}} = {bf{b}}$, analogous to scalar subtraction.
Scalar Multiplication {#scalar-multiplication}
Scalar multiplication involves multiplying a vector a by a scalar (real number) $lambda$ to produce a new vector $lambda{bf{a}}$. If $lambda$ is positive, $lambda{bf{a}}$ is a vector in the same direction as a, but its magnitude is scaled by a factor of $lambda$. Essentially, multiplying by $lambda > 1$ stretches a, while $0 < lambda < 1$ compresses it.
If $lambda$ is negative, $lambda{bf{a}}$ points in the opposite direction of a. The magnitude of $lambda{bf{a}}$ is $|lambda|$ times the magnitude of a. Therefore, regardless of the sign of $lambda$, the magnitude of the scaled vector is given by $|| lambda {bf{a}}|| = |lambda| ||{bf{a}}||$.
Scalar multiplication follows several important properties:
- $s({bf{a}} + {bf{b}}) = s{bf{a}} + s{bf{b}}$ (Distributive Law 1)
- $(s+t){bf{a}} = s{bf{a}} + t{bf{a}}$ (Distributive Law 2)
- $1{bf{a}} = {bf{a}}$ (Identity Property)
- $(-1){bf{a}} = -{bf{a}}$ (Negation Property)
- $0{bf{a}} = {bf{0}}$ (Zero Property)
In the last property, ‘0’ on the left is the scalar zero, while 0 on the right is the zero vector.
If ${bf{a}} = lambda{bf{b}}$ for some scalar $lambda$, vectors a and b are said to be parallel vectors. If $lambda$ is negative, they point in opposite directions, sometimes referred to as anti-parallel, although we will primarily use “parallel” in this broader sense.
We have defined vectors and operations like addition, subtraction, and scalar multiplication geometrically, without relying on coordinate systems. This approach offers the advantage of generality, making these concepts applicable regardless of the coordinate system. However, representing vectors using coordinates is often beneficial, as explored further in our page on vectors in Cartesian coordinate systems.
