Scalars
* Definition: A scalar is a quantity that has only magnitude (size or amount).
* Examples:
* Temperature: 25 degrees Celsius
* Speed: 60 miles per hour
* Mass: 5 kilograms
* Time: 3 minutes
* Distance: 10 meters
* Representation: Scalars are usually represented by a single number and a unit.
* Operations: You can perform basic arithmetic operations (addition, subtraction, multiplication, division) with scalars.
Vectors
* Definition: A vector is a quantity that has both magnitude and direction.
* Examples:
* Displacement: 5 meters east
* Velocity: 10 meters per second north
* Force: 20 Newtons downwards
* Acceleration: 9.8 meters per second squared downwards (due to gravity)
* Representation: Vectors are often represented by an arrow where:
* The length of the arrow represents the magnitude.
* The arrow's direction points in the direction of the vector.
* Operations:
* Addition: To add vectors, you place them head-to-tail and draw the resultant vector from the tail of the first to the head of the last.
* Subtraction: Vector subtraction is equivalent to adding the negative of the vector.
* Multiplication: Vectors can be multiplied by scalars (this scales their magnitude).
* Dot Product: A dot product between two vectors gives a scalar value that is related to the angle between them.
* Cross Product: A cross product between two vectors gives a new vector perpendicular to both original vectors.
Key Differences:
* Magnitude and Direction: Scalars have only magnitude, while vectors have both magnitude and direction.
* Representation: Scalars are single numbers, while vectors are often represented by arrows.
* Operations: The operations applicable to vectors are more complex than those for scalars.
In Summary:
Think of it this way:
* Scalar: Tells you *how much*
* Vector: Tells you *how much* and *which way*