#### Horizontal & Vertical Components of Vectors

A vector in space (3-dimensional) can be defined by its x, y, and z coordinates.

Figure 1:

Point P in 2-D is defined by its Cartesian co-ordinates

(x , y) with respect to the origin (0,0)

We will analyze vectors in two dimensions.

Figure2:

The vector **V** maps the point P in terms

of its position from the origin

In a two dimensional frame we need only consider its x and y co-ordinates.

Note: In figure 2, the tail of vector **V **is at the origin and its head ends at point** P**

Any vector in 2-D can be expressed in terms of its resultant. A resultant is obtained by adding a vector's vertical component in the y direction (see figure 3) to its horizontal component in the x direction (see figure 4) .

Figure 3:**V _{y}** is the vertical component of the

vector

**v**

Figure 4:**V _{x}**

**is the horizontal component of the**

vector

**v**

Figure 5 is a diagram showing both the x and y components defining vector** v. **

Vecor **V** is the resultant vector when **Vx** is added to **Vy** from the tail of **Vx** to the head of** Vy**

Figure 5:

the x and y components of** **vector **v**

We can define **V**_{x }and **V _{y} **in terms of

**V**and the angle using trigonometry:

Know your basic trigonometry functions

Rearrange the first two equations to obtain expressions for **Vy** and **Vx **(the **x** and **y** components of vector **v**)**:**

The resultant vector **v** (see figure 5 above) will now be: **v **=** v _{x }+ v_{y}**

**Examples of problems involving vector components:**

1. What are the horizontal and vertical components of an airplane moving at 300 km/h [E65^{0}N]?

Refer to diagram below

Here = 65 ^{0 }and **v** = 300 km/h

*Answer: Vx = 127 km [to the right] ; Vy = 272 km [up] *

Did you get the answer? Click here to see the solution or hint

2. What are the components of a vector 8 cm long with bearings [W15^{0}S]?

*Answer: Vx =7.7 cm [to the left] ; Vy = 2.1 cm[down]*

Did you get the answer? Click here to see the solution or hint