Vectors and Scalars

Most of the motion variables we're used to dealing with are vectors: they have a size and a direction, and can be expressed as some combination of components in two directions. However, not all variables are vectors; sometimes a number is just a "scalar," a number that has a size but no idea of what direction it points in.

Any time that something can tell you both a direction and a distance, it is a vector. Even in linear motion, the velocity and displacements we worked with were really vectors, not scalars, even though they were mere lone numbers. This is becaus ethat displacement or velocity, in the limitted world of the number line, si enough to tell you exactly where to go, how to displace yourself, how to get there from here.

If an ant is moving at a velocity of (4x + 3y)(m / s), this means that every second, its x coordinate increases by 4 and its y coordinate increases by 3. If it started at the origin, at (0, 0), then after 5 seconds, it is at the position (20, 15). To find the new position, we multiply each coordinate of our vector by the time, 5 s, for which we were moving at that velocity.
A vector.
Multiply by -1.Multiply by 2.

Numbers like the time, 5 s, are called "scalars" because they "scale" (grow, resize) a vector. Scalars are plain, ordinary numbers with no "direction" in their life, no higher purpose except to manipulate vectors. The way they do this is called "scalar multiplication:" to multiply a vector by a scalar, simply multiply each coordinate of the vector by the scalar.

This can become interesting when we have negative scalars. If I asked you where the ant was 5s before it was at the origin, I would be asking for the position at time -5s. The scalar now is a negative number, so rather than simply growing the vector, it turns it completely around. Multiplying a vector by -1 simply reverses its direction.

The following problems are similar to those you did in linear motion, except that now you know that velocity, displacement, and acceleration are vectors, and you can deal with them pointing in two-dimensional directions. Your answer in each case should be a vector.
  1. A rock is sitting on the ground and not moving at all. What is its velocity? (answer as a vector!)






  2. A car is traveling along a straight road with a velocity of (2x + 3y)(m / s). What will its displacement be after 4 s?








  3. A marble rolling down a ramp speeds up with an acceleration of (2x + 1y)(m / s2). If it started at rest, what is its velocity 3s later?








  4. Starting from a position of (3, 10), a dog runs at a velocity of (1x - 3y)(m / s) for 3 s. What is his position at the end of this time?








  5. I displace myself by (-12x + 9y)m in 3 s. What is my average velocity?







  6. Challenge Problem: An object travelling with an initial velocity of (4x + 3y)(m / s) accelerates at (-2x + 0y)(m / s2) for 4 s.

    a) What is its velocity at the end of the time?






    b) What is its displacement during that time?








Answers: 1) (0x + 0y)(m / s) 2) (8x + 12y)m 3) (6x + 3y)(m / s) 4) (6, 1) 5) (-4x + 3y)(m / s) 6) a) (-4x + 3y)(m / s) b) (0x + 12y)(m / s)