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Vector

Vector

A quantity, drawn as an arrow, with both direction and magnitude. For example, force and velocity are vectors. If a quantity has magnitude but not direction, it is called a scalar. Temperature, length, and mass are examples of scalars.

 

Arrow vector on coordinate axes; labels show "magnitude = length" along arrow, "argument = angle" at its angle from horizontal.

 

 

See also

Argument of a vector, vector calculus

Key Formula

v⃗=⟨v1,v2⟩∥v⃗∥=v12+v22\vec{v} = \langle v_1, v_2 \rangle \qquad \|\vec{v}\| = \sqrt{v_1^2 + v_2^2}
Where:
  • v⃗\vec{v} = A vector in two dimensions
  • v1v_1 = The horizontal (x) component of the vector
  • v2v_2 = The vertical (y) component of the vector
  • ∥v⃗∥\|\vec{v}\| = The magnitude (length) of the vector

Worked Example

Problem: A vector goes from the origin to the point (3, 4). Find its magnitude and the unit vector in the same direction.
Step 1: Write the vector in component form.
v⃗=⟨3,4⟩\vec{v} = \langle 3, 4 \rangle
Step 2: Calculate the magnitude using the formula.
∥v⃗∥=32+42=9+16=25=5\|\vec{v}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
Step 3: Find the unit vector by dividing each component by the magnitude. A unit vector has magnitude 1 and points in the same direction.
v^=v⃗∥v⃗∥=⟨3,4⟩5=⟨35,45⟩\hat{v} = \frac{\vec{v}}{\|\vec{v}\|} = \frac{\langle 3, 4 \rangle}{5} = \left\langle \frac{3}{5}, \frac{4}{5} \right\rangle
Step 4: Verify the unit vector has magnitude 1.
∥v^∥=(35)2+(45)2=925+1625=1=1✓\left\|\hat{v}\right\| = \sqrt{\left(\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2} = \sqrt{\frac{9}{25} + \frac{16}{25}} = \sqrt{1} = 1 \checkmark
Answer: The magnitude is 5, and the unit vector is ⟨35,45⟩\left\langle \frac{3}{5}, \frac{4}{5} \right\rangle.

Another Example

This example demonstrates vector addition — combining two vectors — rather than analyzing a single vector. It also shows how to find the direction angle.

Problem: Two forces act on an object: F⃗1=⟨6,2⟩\vec{F}_1 = \langle 6, 2 \rangle and F⃗2=⟨−1,3⟩\vec{F}_2 = \langle -1, 3 \rangle. Find the resultant (total) force vector and its magnitude.
Step 1: Add the vectors component by component to find the resultant.
F⃗R=F⃗1+F⃗2=⟨6+(−1),  2+3⟩=⟨5,5⟩\vec{F}_R = \vec{F}_1 + \vec{F}_2 = \langle 6 + (-1),\; 2 + 3 \rangle = \langle 5, 5 \rangle
Step 2: Calculate the magnitude of the resultant force.
∥F⃗R∥=52+52=25+25=50=52≈7.07\|\vec{F}_R\| = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \approx 7.07
Step 3: Find the direction angle of the resultant using the inverse tangent.
θ=tan⁡−1 ⁣(55)=tan⁡−1(1)=45°\theta = \tan^{-1}\!\left(\frac{5}{5}\right) = \tan^{-1}(1) = 45°
Answer: The resultant force is ⟨5,5⟩\langle 5, 5 \rangle with a magnitude of 52≈7.075\sqrt{2} \approx 7.07 at an angle of 45°45° from the positive x-axis.

Frequently Asked Questions

What is the difference between a vector and a scalar?
A scalar is a quantity described by a single number (its magnitude), such as temperature, mass, or speed. A vector has both magnitude and direction — for example, velocity tells you how fast something moves and in which direction. When you say 'the car goes 60 mph,' that's a scalar (speed); when you say 'the car goes 60 mph north,' that's a vector (velocity).
How do you add two vectors?
You add vectors by adding their corresponding components. If a⃗=⟨a1,a2⟩\vec{a} = \langle a_1, a_2 \rangle and b⃗=⟨b1,b2⟩\vec{b} = \langle b_1, b_2 \rangle, then a⃗+b⃗=⟨a1+b1,  a2+b2⟩\vec{a} + \vec{b} = \langle a_1 + b_1,\; a_2 + b_2 \rangle. Geometrically, you place the tail of the second vector at the tip of the first, and the sum runs from the start of the first to the end of the second (the 'tip-to-tail' method).
What is a unit vector?
A unit vector is a vector with a magnitude of exactly 1. You create it by dividing any nonzero vector by its own magnitude: v^=v⃗/∥v⃗∥\hat{v} = \vec{v} / \|\vec{v}\|. Unit vectors are useful for indicating direction without any scaling. The standard unit vectors in 2D are i^=⟨1,0⟩\hat{i} = \langle 1, 0 \rangle and j^=⟨0,1⟩\hat{j} = \langle 0, 1 \rangle.

Vector vs. Scalar

VectorScalar
DefinitionA quantity with both magnitude and directionA quantity with magnitude only
NotationArrow notation v⃗\vec{v} or component form ⟨a,b⟩\langle a, b \rangleA single number, e.g. 55
ExamplesForce, velocity, displacement, accelerationTemperature, mass, speed, time
AdditionAdd corresponding components; direction mattersAdd ordinary numbers
MultiplicationDot product (gives scalar), cross product (gives vector), scalar multiplicationStandard multiplication of numbers

Why It Matters

Vectors appear throughout physics whenever you study forces, motion, or fields — you cannot solve a force-balance problem or describe projectile motion without them. In mathematics, vectors form the foundation of linear algebra, which is essential for computer graphics, machine learning, and engineering. You will encounter vectors in precalculus or physics classes, and they become even more central in multivariable calculus and beyond.

Common Mistakes

Mistake: Adding magnitudes instead of components when combining vectors.
Correction: Two vectors of magnitude 3 and 4 do not always produce a vector of magnitude 7. You must add the x-components and y-components separately, then compute the magnitude of the result. For example, ⟨3,0⟩+⟨0,4⟩=⟨3,4⟩\langle 3, 0 \rangle + \langle 0, 4 \rangle = \langle 3, 4 \rangle, which has magnitude 5, not 7.
Mistake: Confusing a vector with a point.
Correction: The point (3,4)(3, 4) is a location in the plane. The vector ⟨3,4⟩\langle 3, 4 \rangle describes a displacement — it has the same components but represents a direction and distance, not a fixed position. A vector can be placed anywhere; a point cannot.

Related Terms