3.9.1 Introduction to 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

How are vectors represented?

• Vectors are usually represented using an arrow in the direction of movement
• The length of the arrow represents its magnitude
• They are written as lowercase letters either in bold or underlined
• For example a vector from the point O to A will be written a or a
• The vector from the point A to O will be written -a or -a
• If the start and end point of the vector is known, it is written using these points as capital letters with an arrow showing the direction of movement
• For example: or 
• Two vectors are equal only if their corresponding components are equal
• Numerically, vectors are either represented using column vectors or base vectors
• Unless otherwise indicated, you may carry out all working and write your answers in either of these two types of vector notation

What are column vectors?

• Column vectors are where one number is written above the other enclosed in brackets
• In 2-dimensions the top number represents movement in the horizontal direction (right/left) and the bottom number represents movement in the vertical direction (up/down)
• A positive value represents movement in the positive direction (right/up) and a negative value represents movement in the negative direction (left/down)
• For example: The column vector  represents 3 units in the positive horizontal (x) direction (i.e., right) and 2 units in the negative vertical (y) direction (i.e., down)
• In 3-dimensions the top number represents the movement in the x direction (length), the middle number represents movement in the y direction (width) and the bottom number represents the movement in the z direction (depth)
• For example: The column vector  represents 3 units in the positive x direction, 4 units in the negative y direction and 2 units in the positive z direction

What are base vectors?

• Base vectors use i, j and k notation where i, j and k are unit vectors in the positive x, y, and z directions respectively
• This is sometimes also called unit vector notation
• A unit vector has a magnitude of 1
• In 2-dimensions i represents movement in the horizontal direction (right/left) and j represents the movement in the vertical direction (up/down)
• For example: The vector (-4i + 3j) would mean 4 units in the negative horizontal (x) direction (i.e., left) and 3 units in the positive vertical (y) direction (i.e., up)
• In 3-dimensions i represents movement in the x direction (length), j represents movement in the y direction (width) and k represents the movement in the z direction (depth)
• For example: The vector (-4i + 3j ­- k) would mean 4 units in the negative x direction, 3 units in the positive y direction and 1 unit in the negative z direction
• As they are vectors, i, j and k are displayed in bold in textbooks and online but in handwriting they would be underlined (i, j and k)

How do you know if two vectors are parallel?

• Two vectors are parallel if one is a scalar multiple of the other
• This means that all components of the vector have been multiplied by a common constant (scalar)
• Multiplying every component in a vector by a scalar will change the magnitude of the vector but not the direction
• For example: the vectors  and  will have the same direction but the vector b will have twice the magnitude of a
• They are parallel
• If a vector can be factorised by a scalar then it is parallel to any scalar multiple of the factorised vector
• For example: The vector 9i + 6j – 3k can be factorised by the scalar 3 to 3(3i + 2j – k) so the vector 9i + 6j – 3k is parallel to any scalar multiple of 3i + 2j – k
• If a vector is multiplied by a negative scalar its direction will be reversed
• It will still be parallel to the original vector
• Two vectors are parallel if they have the same or reverse direction and equal if they have the same size and direction