Vectors

Motion
definition

Also known as: vector quantities, arrows

Grade 9-12

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Mathematical quantities that possess both a magnitude (size) and a direction, represented graphically as arrows. Most physical quantities in mechanics — velocity, acceleration, force, displacement, momentum — are vectors.

Definition

Mathematical quantities that possess both a magnitude (size) and a direction, represented graphically as arrows.

💡 Intuition

An arrow pointing somewhere with a certain length—the length is 'how much,' the direction is 'which way.'

🎯 Core Idea

Vectors capture directional quantities; scalars (like mass) don't have direction.

Example

Velocity 30 m/s north, Force 10 N downward, Displacement 5 m east.

Notation

\vec{v} denotes a vector, |\vec{v}| or v is its magnitude, v_x and v_y are its components, \hat{i} and \hat{j} are unit vectors along the x- and y-axes, and \theta is the angle measured from the positive x-axis.

🌟 Why It Matters

Most physical quantities in mechanics — velocity, acceleration, force, displacement, momentum — are vectors. Without vector addition, you cannot correctly combine forces, predict trajectories, or navigate. Vectors are equally fundamental in engineering, computer graphics, and robotics.

💭 Hint When Stuck

When working with vectors, resolve each vector into horizontal (x) and vertical (y) components using v_x = v\cos\theta and v_y = v\sin\theta. Add all x-components together and all y-components together. Then find the resultant magnitude with Pythagoras and the direction with inverse tangent.

Formal View

A vector \vec{v} in 2-D is written as \vec{v} = v_x \hat{i} + v_y \hat{j}, where v_x = |\vec{v}|\cos\theta and v_y = |\vec{v}|\sin\theta. The magnitude is |\vec{v}| = \sqrt{v_x^2 + v_y^2} and the direction is \theta = \arctan(v_y/v_x). Vector addition is component-wise: \vec{a} + \vec{b} = (a_x + b_x)\hat{i} + (a_y + b_y)\hat{j}.

Related Concepts

DisplacementVelocityForce

Compare With Similar Concepts

Speed vs Velocity

🚧 Common Stuck Point

Adding vectors isn't like adding numbers—you must account for direction.

⚠️ Common Mistakes

  • Adding vector magnitudes directly — two forces of 3 N and 4 N at right angles produce a resultant of 5 N (not 7 N); you must use vector addition.
  • Forgetting to include direction in the answer — '5 m/s' is a speed (scalar), while '5 m/s north' is a velocity (vector); the direction is essential.
  • Resolving components with the wrong trigonometric function — use cosine for the component adjacent to the angle and sine for the component opposite.

Common Mistakes Guides

Units and Signs

Frequently Asked Questions

What is Vectors in Physics?

Mathematical quantities that possess both a magnitude (size) and a direction, represented graphically as arrows.

When do you use Vectors?

When working with vectors, resolve each vector into horizontal (x) and vertical (y) components using v_x = v\cos\theta and v_y = v\sin\theta. Add all x-components together and all y-components together. Then find the resultant magnitude with Pythagoras and the direction with inverse tangent.

What do students usually get wrong about Vectors?

Adding vectors isn't like adding numbers—you must account for direction.

How Vectors Connects to Other Ideas

Once you have a solid grasp of vectors, you can move on to displacement, velocity and force.

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