# 3. Vectors in 2 Dimensions

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Components of Vectors

Magnitude of a Vector

Direction of a Vector

So far we have considered 1-dimensional vectors only.

Now we extend the concept to vectors in 2-dimensions. We can use the familiar *x-y* coordinate plane to draw our 2-dimensional vectors.

The vector **V** shown above is a 2-dimensional vector drawn on the *x*-*y* plane.

The vector **V** is acting in 2 different directions simultaneously (to the right and in the up direction). We can see that it has an *x*-component (`6` units to the right) and a *y*-component (`3` units up).

## Components of Vectors

Reading from the diagram above, the *x*-component of the vector **V** is `6` units.

The *y*-component of the vector **V** is `3` units.

We can write these vector components using subscripts as follows:

V_{x}= 6 units

V_{y}= 3 units

## Magnitude of a 2-dimensional Vector

The magnitude of a vector is simply the length of the vector. We can use Pythagoras' Theorem to find the length of the vector **V** above.

Recall (from Section 1, Vector Concepts) that we write the magnitude of **V** using the vertical lines notation **| V |**.

We have:

Magnitude of

`V``= | V | ` `= sqrt(6^2+ 3^2)`

`= sqrt(45)`

`= 6.71\ "units"`

## Direction of a 2-dimensional Vector

To describe the direction of the vector, we normally use degrees (or radians) from the horizontal, in an anti-clockwise direction.

We use simple trigonometry to find the angle. In the above example, we know the **opposite** (`3` units) and the **adjacent** (`6` units) values for the angle (*θ*) we need.

So we have:

`tan theta=3/6=0.5`

This gives:

θ= arctan 0.5

= 26.6°

(= 0.464 radians)

So our vector has magnitude 6.71 units and direction 26.6° up from the right horizontal axis.

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