School of Mathematics and Statistics, The University of Sydney
 3. Addition and subtraction of vectors
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The triangle rule for addition

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Another way to define addition of two vectors is by a head-to-tail construction that creates two sides of a triangle. The third side of the triangle determines the sum of the two vectors, as shown below.

Place the tail of the vector v at the head of the vector u. That is, u = - --> OA and v = ---> AP.

                         P                     v            u        A O

Now construct the vector - --> OP to complete the third side of the triangle OAP.

                                          ---> The vector u + v is defined to be the vectorOP  .

                         P              u + v                     v          u                 A O

This method is equivalent to the parallelogram law of addition, as can be easily seen by drawing a copy of v tail-to-tail with u, to obtain the same parallelogram as before.

                         P        B             u + v                     v    v          u        A O

Using position vector notation, the triangle rule of addition is written as follows: for any three points X, Y , Z,

---->    ---->    ---> XZ   = XY  +  YZ.

Both the triangle and the parallelogram rules of addition are procedures that are independent of the order of the vectors; that is, using either rule, it is always true that u + v = v + u for all vectors u and v. This is known as the commutative law of addition. There are other rules like this one, and they are discussed in the component Vector Algebra.

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