Stretching and flipping a vector

In the picture

1. 1Vector u: the starting arrow, 2 squares right and 1 up
2. 22u: same direction and same line, twice as long
3. 3−2u: twice as long, on a parallel line, pointing the opposite way
4. 4The shared line: u and 2u lie along it, so they are collinear

Examples

• A cyclist who doubles their speed on a straight road: the velocity vector keeps its direction and becomes twice as long.
• A car backing up along a street: its velocity points the other way along the same street, like multiplying by a negative number.

Quiz: Stretching and flipping a vector (Age 15)

1. 1

   A vector u is multiplied by −3. How does the length of −3u compare with the length of u?

   • It is 3 times as long
   • It is one third as long
   • It is the same length, just reversed
2. 2

   If k is a negative number, which way does k·u point compared with u?

   • The opposite way to u
   • The same way as u
   • At a right angle to u
3. 3

   Are the vectors u and −5u collinear?

   • Yes, because u and k·u always lie on the same line or on parallel lines
   • No, because they point in opposite directions
   • Only if they lie on exactly the same line

The answers, each with its explanation, are in the app. This lesson has left the free week, so its quiz is part of the subscription.