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Adding Vectors Using the Head-to-Tail Method

Today's discussion revolves around the process of adding two, three or more vectors through the head-to-tail technique, a fundamental concept in linear algebra.

To provide a clearer explanation of this method, I find it most effective to illustrate it through a hands-on example.

Begin by plotting three vectors - u, v, and w - on a Cartesian plane.

example of three vectors

Choose any one of them and align its tail at the origin point (0,0).

Let's work with vector u as an example.

choose a vector

Now comes the implementation of the head-to-tail strategy.

Select another vector, in this case, v, and align its tail with the head of the vector previously moved to the origin.

select an another vector

Next in line is the vector w.

Position it such that its tail aligns with the head of the recently moved vector, thus completing a sequential arrangement of vectors from the origin.

move last vector

Subsequently, sketch a new vector originating from the origin (0,0) and terminating at the head of the last vector in this sequence.

This newly drawn vector, z, is the summation of our initial three vectors.

the sum of vectors

Hence, we represent the vector sum as

$$ z = u+v+w $$

This method offers an intuitive and geometric approach to adding three or more vectors, achieved by aligning them sequentially in a head-to-tail manner, beginning at the origin.

Note. The sequence of vector selection isn't critical to the outcome. Regardless of the order you select, the final result remains unchanged. To illustrate, if you opt for vector v initially (v2), followed by vector w (named w2), and then conclude with vector u (named u2), the sum vector attained is invariably identical (z).
sum addition




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