Introduction to Vectors (DP IB Applications & Interpretation (AI)): Revision Note

Scalars & Vectors

What are scalars?

• Scalars are quantities without direction

  • They have only a size (magnitude)

  • For example: speed, distance, time, mass

• Most scalar quantities can never be negative

  • You cannot have a negative speed or distance

What are vectors?

• Vectors are quantities which also have a direction, this is what makes them more than just a scalar

  • For example: two objects with velocities of 7 m/s and ‑7 m/s are travelling at the same speed but in opposite directions

• A vector quantity is described by both its magnitude and direction

• A vector has components in the direction of the x- , y-, and z- axes

  • Vector quantities can have positive or negative components

• Some examples of vector quantities you may come across are displacement, velocity, acceleration, force/weight, momentum

  • Displacement is the position of an object from a starting point

  • Velocity is a speed in a given direction (displacement over time)

  • Acceleration is the change in velocity over time

• Vectors may be given in either 2- or 3- dimensions

Vector Notation

How are vectors represented?

• Vectors are usually represented using an arrow in the direction of movement

  • The length of the arrow usually represents its magnitude

• They are written as lowercase letters either in bold or underlined

  • For example, a vector from the point O to A can be written a or a

• You can denote a vector in terms of its start and end points

  • For example, is the vector from point to point

• Two vectors are equal only if their corresponding components are equal

What are base vectors?

• Base vectors use i, j and k notation

  • is a vector of magnitude 1 in the positive direction

  • is a vector of magnitude 1 in the positive direction

  • is a vector of magnitude 1 in the positive direction

• Any vector in 2D or 3D can be written in terms of the base vectors

  • For example, describes the vector that moves

    • 2 in the positive direction

    • 3 in the negative direction

    • 1 in the positive direction

What are column vectors?

• Column vectors list the components of a vector in a column

• The base vectors can be written as column vectors

  • 

  • 

  • 

• Any vector in 2D or 3D can be written as a column vector

  •