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 Ax and Ay respectively. Ax is the x-component and Ay 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”.

Resolution of vector into components\({A}_{x}\hat {x}\) and \({A}_{y}\hat {y}\) are vectors advance from origin to point Ax  and Ay 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:

3D Vector Resolution in Physics

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

 

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:

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

 

Additional Resources