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Components of a Vector

In a two-dimensional coordinate system, any vector can be broken into x -component and y -component.

v = v x , v y

For example, in the figure shown below, the vector v is broken into two components, v x and v y . Let the angle between the vector and its x -component be θ .

The vector and its components form a right angled triangle as shown below.

In the above figure, the components can be quickly read. The vector in the component form is v = 4 , 5 .

The trigonometric ratios give the relation between magnitude of the vector and the components of the vector.

cos θ = Adjacent Side Hypotenuse = v x v

sin θ = Opposite Side Hypotenuse = v y v

v x = v cos θ

v y = v sin θ

Using the Pythagorean Theorem in the right triangle with lengths v x and v y :

| v | = v x 2 + v y 2

Here, the numbers shown are the magnitudes of the vectors.

Case 1: Given components of a vector, find the magnitude and direction of the vector.

Use the following formulas in this case.

Magnitude of the vector is | v | = v x 2 + v y 2 .

To find direction of the vector, solve tan θ = v y v x for θ .

Case 2: Given the magnitude and direction of a vector, find the components of the vector.

Use the following formulas in this case.

v x = v cos θ

v y = v sin θ

Example:

The magnitude of a vector F is 10 units and the direction of the vector is 60 ° with the horizontal. Find the components of the vector.

F x = F cos 60 ° = 10 1 2 = 5

F y = F sin 60 ° = 10 3 2 = 5 3

So, the vector F is 5 , 5 3 .