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In the figure shown below, four vectors namely a , b , c and d are combined to give their sum. Starting with any vector, we add vectors in a manner that the subsequent vector begins at the arrow end of the preceding vector. The illustrations in figures i, iii and iv begin with vectors a , d and c respectively.

Matter of fact, polygon formation has great deal of flexibility. It may appear that we should elect vectors in increasing or decreasing order of direction (i.e. the angle the vector makes with reference to the direction of the first vector). But, this is not so. This point is demonstrated in figure (i) and (ii), in which the vectors b and c have simply been exchanged in their positions in the sequence without affecting the end result.

It means that the order of grouping of vectors for addition has no consequence on the result. This characteristic of vector addition is known as “associative” property of vector addition and is expressed mathematically as :

$\begin{array}{c}\left(\mathbf{a}+\mathbf{b}\right)+\mathbf{c}=\mathbf{a}+\left(\mathbf{b}+\mathbf{c}\right)\end{array}$

Subtraction

Subtraction is considered an addition process with one modification that the second vector (to be subtracted) is first reversed in direction and is then added to the first vector. To illustrate the process, let us consider the problem of subtracting vector, b , from , a . Using graphical techniques, we first reverse the direction of vector, b , and obtain the sum applying triangle or parallelogram law.

Symbolically,

$\begin{array}{l}\mathbf{a}-\mathbf{b}=\mathbf{a}+\left(-\mathbf{b}\right)\end{array}$

Similarly, we can implement subtraction using algebraic method by reversing sign of the vector being subtracted.

Vector method requires that all vectors be drawn true to the scale of magnitude and direction. The inherent limitation of the medium of drawing and measurement techniques, however, renders graphical method inaccurate. Analytical method, based on geometry, provides a solution in this regard. It allows us to accurately determine the sum or the resultant of the addition, provided accurate values of magnitudes and angles are supplied.

Here, we shall analyze vector addition in the form of triangle law to obtain the magnitude of the sum of the two vectors. Let P and Q be the two vectors to be added, which make an angle θ with each other. We arrange the vectors in such a manner that two adjacent sides OA and AB of the triangle OAB, represent two vectors P and Q respectively as shown in the figure.

According to triangle law, the closing side OB represent sum of the vectors in both magnitude and direction.

$\begin{array}{c}\mathbf{OB}=\mathbf{OA}+\mathbf{AB}=\mathbf{P}+\mathbf{Q}\end{array}$

In order to determine the magnitude, we drop a perpendicular BC on the extended line OC.

In ∆ACB,

$\begin{array}{l}\mathbf{AC}=\mathbf{AB}cos\theta =\mathbf{Q}cos\theta \\ \mathbf{BC}=\mathbf{AB}sin\theta =\mathbf{Q}sin\theta \end{array}$

In right ∆OCB, we have :

$\begin{array}{c}\mathrm{OB}=\surd \left({\mathrm{OC}}^{2}+{\mathrm{BC}}^{2}\right)=\surd \left\{{\left(\mathrm{OA}+\mathrm{AC}\right)}^{2}+{\mathrm{BC}}^{2}\right\}\end{array}$

Substituting for AC and BC,

$\begin{array}{l}⇒\mathrm{OB}=\surd \left({\left(P+Qcos\theta \right)}^{2}+{Qsin\theta }^{2}\right)\\ ⇒\mathrm{OB}=\surd \left({P}^{2}+{Q}^{2}{cos}^{2}\theta +2PQcos\theta +{Q}^{2}{sin}^{2}\theta \right)\\ ⇒R=\mathrm{OB}=\surd \left({P}^{2}+2PQcos\theta +{Q}^{2}\right)\end{array}$

Let "α" be the angle that line OA makes with OC, then

$\begin{array}{l}\mathrm{tan}\alpha =\frac{\mathrm{BC}}{\mathrm{OC}}=\frac{Qsin\theta }{P+Qcos\theta }\end{array}$

The equations give the magnitude and direction of the sum of the vectors. The above equation reduces to a simpler form, when two vectors are perpendicular to each other. In that case, θ = 90°; sinθ = sin90° = 1; cosθ = cos90° = 0 and,

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