Vector addition

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, \vec A and \vec B, which are in the same direction. We can add them together by adding their magnitudes together.

For example, if \vec A has a magnitude of 3m and \vec B has a magnitude of 5m, then the resultant vector \vec R will have a magnitude of 3m+5m=8m.

That’s because, if you suppose the vector represents a movement, then we will have moved 3m in the direction of \vec A, and then 5m in the same direction, so we will have moved a total of 8m in that direction.

A car moves 30km north, then 20km north. What is the resultant vector for the car’s movement?

• The car moves 30km north, then 20km north, so the car in total moves 30km+20km=50km.
• We’re still going North, so the direction of the resultant vector is north.
• Answer: 50km north

Adding vectors of opposite directions

If we have two vectors, \vec A and \vec B, 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 \vec A has a magnitude of 3m and \vec B has a magnitude of 5m, and \vec A is in the opposite direction to \vec B.
• the resultant vector \vec R will have a magnitude of 5m-3m=2m, and the direction of \vec R will be the same as the direction of \vec B (because \vec B has the larger magnitude).
• Answer: 2m in the same direction as \vec B

A car moves 30km north, then 20km south. What is the resultant vector for the car’s movement?

• The car moves 30km north, then 20km south, so the car in total moves 30km-20km=10km.
• 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: 10km north

A person cycles 10km east, then 15km west. What is the resultant vector for the person’s movement?

• The person cycles 10km east, then 15km west, so the person in total moves 15km-10km=5km.
• 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: 5km 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 90\degree, then it is pointing east, because it is 90\degree 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 6km east of a lighthouse, and boat B is 8km north of the lighthouse. What is the resultant position vector OF B FROM A?

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:
  • a^2+b^2=c^2
  • 6^2+8^2=c^2
  • 36+64=c^2
  • 100=c^2
  • c=10km
• The direction is the angle \theta that the hypotenuse makes with the east direction, so:
  • \tan\theta=\frac{8}{6}
  • \theta=\tan^{-1}\frac{8}{6}
  • \theta\approx53.13\degree
  • To make it into a bearing north, add the 270\degree that we need to rotate from north to the west line:
    • \theta=270\degree+53.13\degree=323.13\degree
• Answer: 10km at a bearing of 323.13\degree