A vector is a quantity that has both a size, called its magnitude, and a direction. It is drawn as an arrow from an initial point to a terminal point and written in components, such as v = ⟨ 4, 3 ⟩: 4 units right and 3 units up. Velocity and force are vectors; speed and mass are not.
In Spanish
vector
Also called
directed line segment, Euclidean vector, displacement
Subtract the initial point from the terminal point to get the components ⟨ 4, 3 ⟩. The components are the legs of a right triangle, so the magnitude is 5.
Formal definition and notation
A vector in the plane is a directed line segment, where two directed segments with the same length and direction represent the same vector. With initial point (x1, y1) and terminal point (x2, y2), its component form is ⟨ x2 − x1, y2 − y1 ⟩. The magnitude of v = ⟨ a, b ⟩ is ‖v‖ = √a2 + b2. Vectors add component by component, and a scalar k multiplies each component.
Symbol
Read it as
Meaning
v
vector v
a vector named by a bold letter (handwritten with an arrow over it)
⟨ a, b ⟩
the vector a, b
component form: a across, b up; angle brackets separate it from the point (a, b)
‖v‖ or |v|
the magnitude of v
the length of the arrow
v = ai + bj
a i plus b j
the same vector written with the unit vectors i = ⟨ 1, 0 ⟩ and j = ⟨ 0, 1 ⟩
Key facts
A scalar has size only (5 kilograms, 60 miles per hour). A vector has size and direction (60 miles per hour heading north).
Moving an arrow without turning or stretching it gives the same vector: ⟨ 4, 3 ⟩ from (1, 1) and ⟨ 4, 3 ⟩ from (−2, 0) are equal.
The magnitude of a sum is usually not the sum of the magnitudes: ⟨ 3, 0 ⟩ + ⟨ 0, 4 ⟩ = ⟨ 3, 4 ⟩ has magnitude 5, not 7.
The zero vector ⟨ 0, 0 ⟩ has magnitude 0 and no direction. The vector −v has the same length as v but points the opposite way.
Vector formulas
Vector formulas
v = ⟨x2 − x1, y2 − y1⟩ ‖v‖ = √a2 + b2
(x1, y1)
the initial point (tail)
(x2, y2)
the terminal point (head)
a, b
the horizontal and vertical components of the vector
Add vectors by adding matching components: ⟨ 1, 4 ⟩ + ⟨ 3, −2 ⟩ = ⟨ 4, 2 ⟩. Multiply by a scalar by multiplying each component: 3⟨ 2, −1 ⟩ = ⟨ 6, −3 ⟩.
Vector examples, step by step
Example 1: Component form from two points
Warm-up
Find the component form of the vector from A(2, −1) to B(−3, 5).
Subtract the x-coordinates, terminal minus initial: −3 − 2 = −5.
Subtract the y-coordinates: 5 − (−1) = 6.
Write the components in angle brackets: ⟨ −5, 6 ⟩.
Answer: The vector is ⟨ −5, 6 ⟩: 5 units left and 6 units up.
Example 2: Add vectors and find the magnitude
Core
Let u = ⟨ 2, 7 ⟩ and w = ⟨ 4, −15 ⟩. Find u + w and its magnitude.
Add the x-components: 2 + 4 = 6.
Add the y-components: 7 + (−15) = −8.
So u + w = ⟨ 6, −8 ⟩.
Magnitude: √62 + (−8)2 = √100, which is 10.
Answer:u + w = ⟨ 6, −8 ⟩, with magnitude 10.
Example 3: Resultant velocity
Test-style
A boat heads straight across a river at 12 miles per hour while the current pushes it downstream at 5 miles per hour, at a right angle to the boat’s heading. What is the boat’s actual speed?
Write each velocity as a vector: boat ⟨ 0, 12 ⟩, current ⟨ 5, 0 ⟩.
Add them: ⟨ 0 + 5, 12 + 0 ⟩ = ⟨ 5, 12 ⟩.
The actual speed is the magnitude: √52 + 122 = √169.
√169 = 13. Note it is not 12 + 5 = 17.
Answer: The boat’s actual speed is 13 miles per hour.
Common mistakes
✗ MistakeFrom A(2, −1) to B(−3, 5): initial minus terminal, ⟨ 5, −6 ⟩.
A line segment has two endpoints but no direction: segment AB is the same as segment BA. A vector from A to B is the opposite of the vector from B to A.
An ordered pair (4, 3) names one point. A vector ⟨ 4, 3 ⟩ names a movement that can start at any point.
Quick check: 3 questions
Question 1 of 3
What is the magnitude of ⟨ −9, 12 ⟩?
Show answer and explanation
Answer: C.√(−9)2 + 122 = √225, which is 15. 3 adds the components, 21 adds their sizes, and 225 forgets the square root.
Question 2 of 3
Which quantity is a vector?
Show answer and explanation
Answer: B. The wind has a size (20 miles per hour) and a direction (from the west), so it is a vector. Temperature, mass and time have size only; they are scalars.
Question 3 of 3
Find 2⟨ 3, −1 ⟩ − ⟨ 1, 4 ⟩.
Show answer and explanation
Answer: A.2⟨ 3, −1 ⟩ = ⟨ 6, −2 ⟩. Then ⟨ 6 − 1, −2 − 4 ⟩ = ⟨ 5, −6 ⟩. ⟨ 7, 2 ⟩ adds instead of subtracting, ⟨ 2, −5 ⟩ forgets to double, and ⟨ 5, 2 ⟩ has a sign slip in the second component.
Vector: frequently asked questions
What is the difference between a vector and a scalar?
A scalar is just a number with a unit, such as 50 miles per hour or 8 kilograms. A vector also has a direction, such as 50 miles per hour heading east. Speed is a scalar and velocity is a vector; distance is a scalar and displacement is a vector. Multiplying a vector by a scalar stretches or shrinks it.
How do you find the magnitude of a vector?
Square each component, add the squares, and take the square root: for ⟨ a, b ⟩, the magnitude is √a2 + b2. This is the Pythagorean theorem, because the two components form the legs of a right triangle and the vector is the hypotenuse. For ⟨ 6, 8 ⟩, the magnitude is √100 = 10.
Why are vectors written with angle brackets?
Angle brackets, as in ⟨ 4, 3 ⟩, show that the numbers describe a movement, not a location. Parentheses, as in (4, 3), name a single point. Many textbooks use this convention so a vector and a point with the same numbers are not confused. You will also see vectors written as 4i + 3j.
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