**ANALYTIC METHOD, VECTORS ADDITION.**

**1.- COMPONENTS**

The graphical sum often has not enough exactitude
and is not useful when the **vectors** are in **three dimensions**.
As every vector can be represented as the sum of two other vectors, these
vectors are called the **components** of the original vector. Usually the
components are chosen along two mutually perpendicular directions. For example,
assume the vector **V** below in the figure. It can be split in the component
**V**_{x} parallel to the x axis and the component V_{y}
parallel to the y axis.

We use coordinate axis x-y with origin at the
tail of vector **V**. Notice that **V** = **V**_{x} + **V**_{y}
according to the parallelogram rule.

The magnitudes of **V**_{x} and
**V**_{y} are denoted V_{x} and V_{y}, and are
numbers, positive or negatives as they point at the positive or negative side
of the x-y axis.

Notice also that V_{x} = Vcos
and V_{y} = Vsen.

**2.- UNIT VECTORS**

Vector quantities can often be expressed in
terms of unit vectors. A unit vector is a vector whose magnitude is equal
to one and dimensionless. They are used to specify a determinated direction.
The symbols **i**, **j** y **k** represent unit vectors pointing
in the directions x, y and z positives, respectively.

Now **V** can be written **V** = V_{x}**i**
+ V_{y}**j**.

If we need to add the vector **A** = A_{x}
**i** + A_{y} **j** with

the vector **B** = B_{x} i + B_{y} **j** we write

**R** = **A** + **B** = A_{x} **i** + A_{y} **j**
+ B_{x} **i** + B_{y} **j** = (A_{x} + B_{x})**i**
+ (A_{y} + B_{y})**j**.

The components of **R** are R_{x}
= A_{x} + B_{x} and R_{y} = A_{y} + B_{y}

**Exercise, Example:** Use
of **components** and **unit vectors**.

A boyscout walks 22 km in North direction, and then he walks in direction
60º Southeast during 47.0 km. Find the components of the resulting vector
displacement from the starting point, its magnitude and angle with the x axis.

**Solution:** The two displacements are
shown in the figure, where we choose the positive x axis pointing to East
and the positive y axis pointing to North.

The resultant displacement **D** is the
sum of **D**_{1} and **D**_{2}.

Using unit vectors:

**D**_{1} = 22 **j**

**D**_{2} = 47cos60º i - 47sen60º **j**

Then **D** = **D**_{1}+**D**_{2} = 22 **j** +
47cos60º **i** - 47sen60º **j** = 23.5 **i** - 18.7 **j**

and the resultant vector is completely specified with an x component D_{x}
= 23.5 km and a y component D_{y} = -18.7 km. (Note D_{x}
and D_{y} are scalars).

The same resultant vector can be specified
giving its magnitude and angle:

D^{2} = D_{x}^{2} + D_{y}^{2} = (23.5
km)^{2} + (-18.7 km)^{2} finding D = 30 km.

tan = D_{y}/D_{x}
= -18.7/23.5 = -0.796 finding
= -38.5º (under the x axis) or 38.5 Southeast.

**More on This Theme:**

**
· Vectors, Scalars: Initial Page
· Vectors, Addition Tools, Problems**

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