**RESOLUTION OF VECTOR INTO COMPONENTS IN PHYSICS**

Resolution of a vector into components can be done along each coordinate axis in physics. Suppose a vector \({ \vec { A } } \) lies in a particular plane. This vector can be represented as the sum of two vectors and these two vectors are called components of \({ \vec { A } } \). If \({ \vec { A } } \) is projected onto the perpendicular x and y axes, the projected scalar components of vector \({ \vec { A } } \) are A_{x}and A

_{y}respectively. A

_{x}is the x-component and A

_{y}is the y-component of the vector \({ \vec { A } } \). These scalar components can be positive, zero, or negative.

We can rewrite vector \({ \vec { A } } \) in terms of the two components.

\(\vec {A} ={A}_{x}\hat {x} +{A}_{y}\hat {y}\) Here, \(\hat {x}\) and \(\hat {y}\) are unit vectors. Unit vectors are always pointing in the direction of positive axis. The length of unit vector is 1. Unit vector is denoted with circumflex, or “hat”.\({A}_{x}\hat {x}\) and \({A}_{y}\hat {y}\) are vectors advance from origin to point A_{x } and A_{y} respectively. So these vectors are drawn in the figure.

When these vectors are added together, the result is exactly identical to vector \({ \vec { A } } \). This process is called resolution of a vector into two directions.

The magnitude of the vector \({ \vec { A } } \) is:

\(\left| A \right| =\sqrt { { { A }_{ x } }^{ 2 }+{ { A }_{ y } }^{ 2 } }\) Following trigonometric functions are used to find the components of the vector. \({A}_{x}=A\cos {\theta}\) \({A}_{y}=A\sin {\theta}\) \(\tan{\theta} =\frac{{A}_{y}}{{A}_{x}}\)Here θ is the angle between vector \({ \vec { A } } \) and the positive x-axis. Suppose that the vector \({ \vec { A } } \) lies along the +x-axis and then rotated from the +x-axis toward the +y-axis, then θ is positive; if the rotation is from the +x-axis toward the -y-axis then θ is negative.

## 3D Vector Resolution

Space is composed of three dimensions . Resolution of a vector in three dimensions is an extension of the above method. Trigonometric function for 3D vector decompostions are:

\(A=\sqrt {{A}_{x}^{2}+{A}_{y}^{2}+{A}_{z}^{2}}\) \({ A }_{x}=A\cos{{\theta}_{x}}\) \({ A }_{y}=A\cos{{\theta}_{y}}\) \({ A }_{z}=A\cos{{\theta}_{z}}\)

## Example for Resolution of Vector

What is the angle and the magnitude of the vector \({ \vec { A } } \)?

\(\vec{A=}3\hat{x}-5\hat {y}\)The magnitude of the vector is:

\(\left|A\right|=\sqrt{{3}^{2}+{(-5)}^{2}}=\sqrt{34}\)The angle of the vector is:

\(\tan{\theta}=\frac{{A}_{y}}{{A}_{x}}=\frac{-5}{3}\) \(\theta=\arctan{\frac{-5}{3}}=-59.04 deg\)