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
State whether each of the following is a scalar or a vector quantity.
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)
Write the vector using column vector notation.
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
Show that the vectors and are parallel and find the scalar multiple that maps a onto b.