(2) Adding vectors

and what this can mean physically

Since we can think of vectors as displacements or journeys, to add two vectors we just need to find the single displacement which gives the same result as doing the two displacements separately.
The left-hand drawing below shows that the red single vector of p + q is given by doing p followed by q and also by doing q followed by p. Each of the two joined together triangles shows what is called the triangle law of addition.
So, for example, if a boat sets a course to move with velocity p in water flowing with velocity q,then it will actually have a resultant velocity of p + q. The third side of the triangle gives the speed and direction in which it will move.

adding two vectors

The right hand drawing could represent two forces P and Q acting on a particle. Their joint resultant effect is given by the vector P + Q which is the diagonal of the parallelogram. This picture shows what is called the parallelogram law for the addition of forces . This is the same as the triangle law of addition.
We can show the addition of any number of vectors in a similar way by putting them nose to tail and seeing what the final displacement is. This drawing shows that the successive displacements of a, b, c, d and e are equivalent to the single displacement of r. The addition works in exactly the same way if the vectors shown are being placed nose to tail in 3 dimensions.
adding a number of vectors

We have a + b + c + d + e = r.
You can add the vectors in any order you choose. The result is the same.

If the vectors join up so that there is no gap between the tail of the first vector and the head of the last one then their vector sum is zero. So, for example, if the vectors were representing forces acting on a body, then there would be no net force acting on it (though there could still be a turning effect).

It is very easy to see pictorially how the rules work so far. However, in many circumstances, such as when we need to prove that results are true in all situations, we shall have to handle the vectors by using algebra.
To be able to do this, we need to know how to write them in component form.


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