Physics · Ch 2 — Mathematical Methods

Resolution of vectors

2.4

Resolution of vectors

Just as two or more vectors can be combined into a single resultant vector, a single vector can conversely be expressed as the sum of two or more vectors along chosen fixed directions:

\vec V = V_1\hat n_1+V_2\hat n_2+V_3\hat n_3 \qquad \text{--- (2.11)}

where $\hat n_1,\hat n_2,\hat n_3$ are unit vectors along the chosen directions and $V_1, V_2, V_3$ are called the components of $\vec V$ along those directions. This process of splitting a vector into its components is called resolution of the vector. Components can, in principle, be found along any set of directions, but when the chosen directions are mutually perpendicular, the components are called rectangular components.

Rectangular components in two dimensions. Consider a vector $\vec R = \vec{OC}$ starting at the origin of a rectangular coordinate system. Dropping perpendiculars from C to the x-axis (meeting it at A) and to the y-axis (meeting it at B) gives the two rectangular components $R_x$ and $R_y$ of $\vec R$ along the x- and y-axes. By the parallelogram law,

\vec R = \vec R_x+\vec R_y = R_x\hat i+R_y\hat j

If $\theta$ is the angle $\vec R$ makes with the x-axis, then

R_x = R\cos\theta \qquad \text{--- (2.12)}, \qquad R_y = R\sin\theta \qquad \text{--- (2.13)}

Squaring and adding these gives the magnitude of $\vec R$ :

R = \sqrt{R_x^2+R_y^2} \qquad \text{--- (2.14)}

and the direction of $\vec R$ is

$\theta = \tan^{-1}\left(\dfrac{R_y}{R_x}\right) \qquad \text{--- (2.15)}$ …

Figure 2.8Resolution of a vector into rectangular components

What this figure shows. A rectangular (x-y) coordinate system with origin O. A vector $\vec R=\vec{OC}$ is drawn from the origin to point C somewhere in the plane, at angle $\theta$ to the x-axis. Perpendiculars are dropped from C to meet the x-axis at point A and the y-axis at point B, so that $\vec{OA}$ is the rectangular component $\vec R_x$ along the x-axis and $\vec{OB}$ is the rectangular component $\vec R_y$ along the y-axis (the dashed construction lines from C to A and C to B complete a rectangle OACB with OC as its diagonal). The angle $\theta$ between $\vec R$ and the x-axis is marked at O. No numeric coordinate values are pri …

Misc Ex.2.4Unit vector along a given vector

Worked out. Given the vector $\vec V = 3\hat i+4\hat j$ , the problem asks for the unit vector pointing in the same direction as $\vec V$ . The method first computes the magnitude $|\vec V|=\sqrt{3^2+4^2}=5$ , then divides the vector by its own magnitude, $\hat V = \vec V/|\vec V|$ , to obtain a vector of magnitude exactly 1 pointing the same way — a direct, concrete application of the general unit-vector de …

Misc Ex.2.5Comparing magnitudes of two vectors

Worked out. Given two vectors $\vec a = \hat i+2\hat j$ and $\vec b = 2\hat i+\hat j$ , the problem asks for their magnitudes, and whether the two vectors are equal. The method computes $|\vec a|=\sqrt{1^2+2^2}=\sqrt5$ and $|\vec b|=\sqrt{2^2+1^2}=\sqrt5$ — the magnitudes turn out to be equal — but then checks the EQUALITY-OF-VECTORS condition (corresponding components must match, $a_x=b_x$ and $a_y=b_y$ ), finding $a_x\ne b_x$ and $a_y\ne b_y$ , so despite having equal magnitude the two vectors are NOT …

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