Adding vector quantities

If we have two vector quantities, we can add them to get a new vector quantity.

The thing we get after adding two vectors together is called the resultant, or more specifically, resultant vector.

Adding vectors of the same direction

Suppose we have two vectors, and , which are in the same direction. We can add them together by adding their magnitudes together.

For example, if has a magnitude of and has a magnitude of , then the resultant vector will have a magnitude of .

That’s because, if you suppose the vector represents a movement, then we will have moved in the direction of , and then in the same direction, so we will have moved a total of in that direction.

A car moves north, then north. What is the resultant vector for the car’s movement?

• The car moves north, then north, so the car in total moves .
• We’re still going North, so the direction of the resultant vector is north.
• Answer: north

Adding vectors of opposite directions

If we have two vectors, and , which are in opposite directions, then we can add them together by subtracting the smaller magnitude from the larger magnitude, and keeping the direction of the larger vector.

• Let’s suppose has a magnitude of and has a magnitude of , and is in the opposite direction to .
• the resultant vector will have a magnitude of , and the direction of will be the same as the direction of (because has the larger magnitude).
• Answer: in the same direction as

A car moves north, then south. What is the resultant vector for the car’s movement?

• The car moves north, then south, so the car in total moves .
• The car is still moving north, because the north movement is larger than the south movement, so the direction of the resultant vector is north.
• Answer: north

A person cycles east, then west. What is the resultant vector for the person’s movement?

• The person cycles east, then west, so the person in total moves .
• The person is still moving west, because the west movement is larger than the east movement, so the direction of the resultant vector is west.
• Answer: west

Bearings

We can write the direction of a vector as a bearing, which is the angle that the vector makes with the north direction, measured clockwise.

For example, if a vector has a bearing of , then it is pointing east, because it is clockwise from north.

Adding vectors at right angles

We can also find the resultant of two vectors if they’re at right angles to each other.

We can use:

• the Pythagorean theorem to find the magnitude of the resultant vector
• trigonometry to find the direction of the resultant vector

Boat A is east of a lighthouse, and boat B is north of the lighthouse. What is the resultant position vector OF FROM ?

Draw a right triangle with the lighthouse at the right angle, boat A at one end of the hypotenuse, and boat B at the other end of the hypotenuse:

        B
          +
          | \
          |   \
          |     \
          |       \
     8km  |         \
          |           \
          |             \
          |            θ  \
          + --------------- +
        L         6km         A

• The magnitude is the hypotenuse of the triangle, so:
• The direction is the angle that the hypotenuse makes with the east direction, so:
  • To make it into a bearing north, add the that we need to rotate from north to the west line:
• Answer: at a bearing of