How To Find The X Component Of A Vector

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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.

$\vec{v} = ⟨ v_{x}, v_{y} ⟩$

For example, in the figure shown below, the vector $\vec{v}$ 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 $\vec{v} = ⟨ 4, 5 ⟩$ .

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

$\cos θ = \frac{Adjacent Side}{Hypotenuse} = \frac{5_{x}}{v}$

$\sin θ = \frac{Opposite Side}{Hypotenuse} = \frac{v_{y}}{five}$

$v_{ten} = v \cos θ$

$v_{y} = v \sin θ$

Using the Pythagorean Theorem in the correct triangle with lengths $v_{x}$ and $v_{y}$ :

$| v | = \sqrt{v_{10}^{2} + v_{y}^{2}}$

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 $| five | = \sqrt{5_{x}^{2} + v_{y}^{2}}$ .

To find direction of the vector, solve $\tan θ = \frac{v_{y}}{v_{ten}}$ 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 $\vec{F}$ is $10$ units and the direction of the vector is $lx °$ with the horizontal. Discover the components of the vector.