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
• 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)

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
  • 
    • 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