Question

The component of a vector is.

Answer 1

Answer:

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

v⃗ =⟨vx,vy⟩

For example, in the figure shown below, the vector v⃗ is broken into two components, vx and vy . 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 v⃗ =⟨4,5⟩ .

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

cosθ=Adjacent SideHypotenuse=vxv

sinθ=Opposite SideHypotenuse=vyv

vx=vcosθ

vy=vsinθ

Using the Pythagorean Theorem in the right triangle with lengths vx and vy :

| v |=vx2+vy2−−−−−−−√

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 |=vx2+vy2−−−−−−−√ .

To find direction of the vector, solve tanθ=vyvx 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.

vx=vcosθ

vy=vsinθ

Explanation:

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