26 CHAPTER 2. Fn

Vectors are used to model force and other physical vectors like velocity. What was justdescribed would be called a force vector. It has two essential ingredients, its magnitudeand its direction.

Note there are n special vectors which point along the coordinate axes. These are

ei ≡ (0, · · · ,0,1,0, · · · ,0)

where the 1 is in the ith slot and there are zeros in all the other spaces. See the picture inthe case of R3.

ye2

z

e3

x

e1

The direction of ei is referred to as the ith direction. Given a vector v = (a1, · · · ,an) , itfollows that

v = a1e1 + · · ·+anen =n

∑k=1

aiei.

What does addition of vectors mean physically? Suppose two forces are applied tosome object. Each of these would be represented by a force vector and the two forcesacting together would yield an overall force acting on the object which would also be a forcevector known as the resultant. Suppose the two vectors are a =∑

nk=1 aiei and b =∑

nk=1 biei.

Then the vector a involves a component in the ith direction, aiei while the component inthe ith direction of b is biei. Then it seems physically reasonable that the resultant vectorshould have a component in the ith direction equal to (ai +bi)ei. This is exactly what isobtained when the vectors, a and b are added.

a+b = (a1 +b1, · · · ,an +bn) =n

∑i=1

(ai +bi)ei.

Thus the addition of vectors according to the rules of addition in Rn which were pre-sented earlier, yields the appropriate vector which duplicates the cumulative effect of allthe vectors in the sum.

An item of notation should be mentioned here. In the case of Rn where n ≤ 3, it isstandard notation to use i for e1, j for e2, and k for e3. Now here are some applications ofvector addition to some problems.

Example 2.8.2 There are three ropes attached to a car and three people pull on theseropes. The first exerts a force of 2i+3j−2k Newtons, the second exerts a force of 3i+5j+kNewtons and the third exerts a force of 5i− j+2k. Newtons. Find the total force in thedirection of i.

To find the total force add the vectors as described above. This gives 10i+7j+kNewtons. Therefore, the force in the i direction is 10 Newtons.

As mentioned earlier, the Newton is a unit of force like pounds.