Scalars and vectors.

Step 1 of 6

A size with a direction

Some quantities are fully described by a number and a unit: a mass of 3 kg, a time of 10 s, a temperature of 20 °C. These are scalars. Others need a direction too: knowing that you walked 5 m is not enough to know where you are; you also need to know which way. These are vectors.

A vector is drawn as an arrow: its length is the size, and the way it points is the direction. On the board, the arrow $u$⁠ is a walk of 5 m, and each grid square is a metre. The numbers under the board split the arrow into a sideways step and an upward step; you will come back to them in the Unit on trigonometry.

Step 2 of 6

Drawing an arrow

To describe a vector you need two things: the size, that is the length of the arrow, and the direction. Here the direction is measured in degrees from the direction to the right, anticlockwise: $0^\circ$⁠ right, $90^\circ$⁠ up, $180^\circ$⁠ left and $270^\circ$⁠ down. Take right to be east and up to be north, as on a map.

Change Length of u and Direction of u.

Step 3 of 6

Speed and velocity

The arrows $u$⁠ and $v$⁠ on the board are two cyclists setting off from the same point, and each arrow shows where its cyclist gets to in one second: 3 m north, and 3 m south.

In everyday language they ride “at the same speed”. Physics tells the two apart: speed, how fast, is a scalar. Velocity, how fast and which way, is a vector.

Step 4 of 6

There and back

Distance is how far you walked in all, along the whole path: a scalar. Displacement is the arrow from the starting point to the end point: a vector. When you walk in a straight line in one direction they are equal in size, but not always.

You walk 5 m east, $u$⁠, then 5 m west, $v$⁠. Answer, then choose Added under Look inside: the arrow $v$⁠ moves to start where $u$⁠ ends, and the arrow $u + v$⁠, from start to finish, is the displacement.

Step 5 of 6

Along a line

When every move is along the same line, you add them with signs: east positive and west negative. A walk of 6 m east followed by 2 m west is $6 + (-2) = 4$⁠, a displacement of 4 m east, like jumps on the number line.

Here $u$⁠ is 6 m east, and $v$⁠ points west. Change Length of v.

Step 6 of 6

Round the corner

You walk 3 m east, then 4 m north. The distance is 7 m. But the displacement is the straight arrow from start to finish, and it cuts the corner. Predict, then see $u + v$⁠ on the board.