In a two-dimensional coordinate system, any kind of vector have the right to be broken into x -component and y -component.

v → = 〈 v x , v y 〉

because that example, in the figure presented below, the vector v → is broken into two components, v x and v y . Allow the angle between the vector and its x -component it is in θ .

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The vector and its components type a appropriate angled triangle as displayed below.

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In the above figure, the materials can be easily read. The vector in the component type is v → = 〈 4 , 5 〉 .

The trigonometric ratios offer the relation between size of the vector and also the materials of the vector.

cos θ = AdjacentSide Hypotenuse = v x v

sin θ = OppositeSide Hypotenuse = v y v

v x = v cos θ

v y = v sin θ

using the Pythagorean to organize in the appropriate triangle with lengths v x and also v y :

|   v   | = v x 2 + v y 2

Here, the numbers presented are the magnitudes that the vectors.

situation 1: Given contents of a vector, find the magnitude and also direction of the vector.

usage the following formulas in this case.

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

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

case 2: offered the magnitude and also direction of a vector, discover the contents of the vector.

usage the complying with formulas in this case.

v x = v cos θ

v y = v sin θ




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Example:

The magnitude of a vector F → is 10 units and the direction the the vector is 60 ° through the horizontal. Find the contents of the vector.

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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 〉 .