Suppose you push on something. What is important? There are really two things which are important, how hard you push and the direction you push.
Definition 2.8.1 Force is a vector. The magnitude of this vector is a measure of how hard it is pushing. It is measured in units such as Newtons or pounds or tons. Its direction is the direction in which the push is taking place.
Of course this is a little vague and will be left a little vague until the presentation of Newton’s second law later. See the appendix on this or any physics book.
Vectors are used to model force and other physical vectors like velocity. What was just described would be called a force vector. It has two essential ingredients, its magnitude and its direction. Think of vectors as directed line segments or arrows as shown in the following picture in which all the directed line segments are considered to be the same vector because they have the same direction, the direction in which the arrows point, and the same magnitude (length).
Because of this fact that only direction and magnitude are important, it is always possible to put a vector in a certain particularly simple form. Let

where t ∈

The point in ℝ^{n},q − p, will represent the vector. Geometrically, the arrow
Definition 2.8.2 Let x =
It is customary to identify the point in ℝ^{n} with its position vector.
The magnitude of a vector determined by a directed line segment

and for v any vector in ℝ^{n} the magnitude of v equals
Example 2.8.3 Consider the vector v ≡

What is the geometric significance of scalar multiplication? If a represents the vector v in the sense that when it is slid to place its tail at the origin, the element of ℝ^{n} at its point is a, what is rv?
= ^{1∕2} =
^{1∕2}


= ^{1∕2}
^{1∕2} =
. 
Thus the magnitude of rv equals
Note there are n special vectors which point along the coordinate axes. These are

where the 1 is in the i^{th} slot and there are zeros in all the other spaces. See the picture in the case of ℝ^{3}.
The direction of e_{i} is referred to as the i^{th} direction. Given a vector v =
What does addition of vectors mean physically? Suppose two forces are applied to some object. Each of these would be represented by a force vector and the two forces acting together would yield an overall force acting on the object which would also be a force vector known as the resultant. Suppose the two vectors are a =∑ _{k=1}^{n}a_{i}e_{i} and b =∑ _{k=1}^{n}b_{i}e_{i}. Then the vector a involves a component in the i^{th} direction, a_{i}e_{i} while the component in the i^{th} direction of b is b_{i}e_{i}. Then it seems physically reasonable that the resultant vector should have a component in the i^{th} direction equal to

Thus the addition of vectors according to the rules of addition in ℝ^{n} which were presented earlier, yields the appropriate vector which duplicates the cumulative effect of all the vectors in the sum.
What is the geometric significance of vector addition? Suppose u,v are vectors

Then u + v =
An item of notation should be mentioned here. In the case of ℝ^{n} where n ≤ 3, it is standard notation to use i for e_{1},j for e_{2}, and k for e_{3}. Now here are some applications of vector addition to some problems.
Example 2.8.4 There are three ropes attached to a car and three people pull on these ropes. The first exerts a force of 2i+3j−2k Newtons, the second exerts a force of 3i+5j + k Newtons and the third exerts a force of 5i − j+2k. Newtons. Find the total force in the direction of i.
To find the total force add the vectors as described above. This gives 10i+7j + k Newtons. Therefore, the force in the i direction is 10 Newtons.
As mentioned earlier, the Newton is a unit of force like pounds.
Example 2.8.5 An airplane flies North East at 100 miles per hour. Write this as a vector.
A picture of this situation follows.
The vector has length 100. Now using that vector as the hypotenuse of a right triangle having equal sides, the sides should be each of length 100/
This example also motivates the concept of velocity.
Definition 2.8.6 The speed of an object is a measure of how fast it is going. It is measured in units of length per unit time. For example, miles per hour, kilometers per minute, feet per second. The velocity is a vector having the speed as the magnitude but also specifying the direction.
Thus the velocity vector in the above example is 100∕
Example 2.8.7 The velocity of an airplane is 100i + j + k measured in kilometers per hour and at a certain instant of time its position is
Consider the vector

Example 2.8.8 A certain river is one half mile wide with a current flowing at 4 miles per hour from East to West. A man swims directly toward the opposite shore from the South bank of the river at a speed of 3 miles per hour. How far down the river does he find himself when he has swam across? How far does he end up swimming?
Consider the following picture.
You should write these vectors in terms of components. The velocity of the swimmer in still water would be 3j while the velocity of the river would be −4i. Therefore, the velocity of the swimmer is −4i + 3j. Since the component of velocity in the direction across the river is 3, it follows the trip takes 1∕6 hour or 10 minutes. The speed at which he travels is
