Figure 1 Coordinate system

Solution: We use Eq.2.8 to find unknown x side

Figure 2.23

Figure 2.24

Unit vectors
A unit vector is a vector that has a magnitude of 1, with no units. Its only purpose is to point-that is, to describe a direction in space. The unit vectors i and j point in the directions of the positive x- and the positive y-axes (Fig.2.23).
Two useful unit vectors are i and j, defined by

Example: Is the vector a unit vector?
Solution: The magnitude of is so this is not a unit vector?
The vector is a unit vector, since its magnitude is 1.



Note that every component of a unit vector must be less than or equal to 1.
Let’s say a vector lies in the xy-plane, as shown in Figure 2.24. Vector has two components, and . We can write a vector in terms of the unit vectors and

A vector (Fig.2.23)in terms of its component is

Figure 2.25

Figure 2.26

Vector addition
When two vectors and are represented in terms of their components, see Figure 2.25, we can express the vector sum using unit vectors as follows:

If the vectors do not all lie in the xy-plane, then we need a third component. We introduce a third unit vector that points in the direction of the positive z-axis, Figure 2.26. There are three special unit vectors, oriented along the three axes. They are called i-hat, j-hat, and k-hat, and they point along the x, y, and z axes, respectively

Then the components of two vectors in three dimensions is




Example 7: (a) What is the sum in unit–vector notation of the two vectors A = 4.0i+ 3.0j and B = −13.0i + 7.0j? (b) What are the magnitude and direction of A + B?