A **vector** is a quantity that has a length (a non-negative real number) as well as direction (or orientation) Vectors can be represented in one, two or three dimensions.

The components of a vector are the projections of said vector on the coordinate axis; In Figure I we see that v_{x }and V_{y} are the projections of the vector **V** on the axes therefore these are the components of

**.**

**V****Defining components of a vector**

Let's consider a rectangular or Cartesian coordinate system (Figure II). The component "x" (which we will call A_{x}) vector ** A** it is the shadow that the latter casts on the x axis; on the other hand, the component "y" (which we will call A

_{y}) vector

**is the shadow that the latter casts on the y axis. The vector sum of both components must result in the vector**

*A***:**

*A*A_{x} + A_{y }= *A*

**Notation**

The components of a vector can be enclosed in parentheses and separated with commas:

** A** = (A

_{x}, TO

_{y})

In the case of three dimensions, it is expressed in this way:

** A** = (A

_{x}, TO

_{y}, TO

_{z})

We can also express them as a combination of unit vectors (i, j, k):

** A** = A

_{x }î + A

_{y }ĵ and

**= A**

*A*_{x }î + A

_{y }ĵ + A

_{z}k

Other times it can be represented in matrix form as:

** A** = [A

_{x}, TO

_{y}, TO

_{z}]

**Rectangular components of a vector**

From Figure II we find that:

- A
_{x}= A cosθ - A
_{y}= A sinθ

These components are the sides of a right triangle whose hypotenuse has a magnitude A.

The module ** A** and its address is related to its components such as:

|** A**| = √ [(A

_{x})

^{2 }+ (A

_{y})

^{2}]

Y

tanθ = A_{x} / TO_{y}

**Exercises**

- Find the magnitude of the vector
= 5î + 3ĵ - 2k*A*

We see that 5 is the "x" component, 3 the "y" component and -2 the "z" component. Now, from the formula that relates the components to the magnitude of the vector, we have:

|** A**| = √ [(A

_{x})

^{2 }+ (A

_{y})

^{2 }+ (A

_{z})

^{2}] = √ [(5)

^{2 }+(3)

^{2 }+ (-2)

^{2}= √ (25 + 9 +4) = 6.16

- Find the direction of the vector
= 4î + 5ĵ*A*

We see that 4 is the component "x" and 5 is "y". Now, from the formula that relates the components to the direction of the vector, we have:

tanθ = A_{x }/ TO_{y} = 4/5

θ = arctan (4/5) = 0.67 rad

- Find the magnitude and direction of the vector
= 9î + 15ĵ*A*

We see that 9 is the component "x" and -15 is "y". From the formula that relates the components to the magnitude of the vector, we have:

|** A**| = √ [(A

_{x})

^{2 }+ (A

_{y})

^{2 }] = √ [(9)

^{2 }+(-15)

^{2}= √ (81 + 225) = 17.49

Now, from the formula that relates the components to the direction of the vector, we have:

tanθ = A_{x }/ TO_{y} = 9/-15

θ = arctan (- 9/15) = - 0.54 rad