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Components of a Vector

In a two-dimensional coordinate organization, whatsoever vector can exist broken into x -component and y -component.

v = v x , v y

For example, in the figure shown below, the vector v is cleaved into ii components, 5 x and v y . Let the angle betwixt the vector and its x -component exist θ .

The vector and its components grade a correct angled triangle as shown beneath.

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

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

cos θ = Adjacent Side Hypotenuse = 5 x v

sin θ = Opposite Side Hypotenuse = v y five

v ten = v cos θ

v y = v sin θ

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

| v | = v 10 2 + v y 2

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

Case ane: Given components of a vector, detect the magnitude and direction of the vector.

Utilize the following formulas in this case.

Magnitude of the vector is | five | = 5 x 2 + v y 2 .

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

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

Use the following formulas in this case.

five x = v cos θ

v y = five sin θ

Example:

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

F x = F cos 60 ° = 10 1 2 = 5

F y = F sin 60 ° = 10 3 two = v iii

So, the vector F is 5 , 5 3 .