1.5 Vector addition  (Page 4/4)

These results for vectors at right angle are exactly same as determined, using Pythagoras theorem.

Nature of vector addition

Vector sum and difference

The magnitude of sum of two vectors is either less than or equal to sum of the magnitudes of individual vectors. Symbolically, if a and b be two vectors, then

$$\begin{array}{c}|\mathbf{a}+\mathbf{b}|\le \left|\mathbf{a}\right|+\left|\mathbf{b}\right|\end{array}$$

We know that vectors a , b and their sum a + b is represented by three side of a triangle OAC. Further we know that a side of triangle is always less than the sum of remaining two sides. It means that :

$$\begin{array}{c}\mathrm{OC}<\mathrm{OA}+\mathrm{AC}\\ \mathrm{OC}<\mathrm{OA}+\mathrm{OB}\\ \Rightarrow |\mathbf{a}+\mathbf{b}|<\left|\mathbf{a}\right|+\left|\mathbf{b}\right|\end{array}$$

There is one possibility, however, that two vectors a and b are collinear and act in the same direction. In that case, magnitude of their resultant will be "equal to" the sum of the magnitudes of individual vector. This magnitude represents the maximum or greatest magnitude of two vectors being combined.

$$\begin{array}{c}\mathrm{OC}=\mathrm{OA}+\mathrm{OB}\\ \Rightarrow |\mathbf{a}+\mathbf{b}|=\left|\mathbf{a}\right|+\left|\mathbf{b}\right|\end{array}$$

Combining two results, we have :

$$\begin{array}{c}|\mathbf{a}+\mathbf{b}|\le \left|\mathbf{a}\right|+\left|\mathbf{b}\right|\end{array}$$

On the other hand, the magnitude of difference of two vectors is either greater than or equal to difference of the magnitudes of individual vectors. Symbolically, if a and b be two vectors, then

$$\begin{array}{c}|\mathbf{a}-\mathbf{b}|\ge \left|\mathbf{a}\right|-\left|\mathbf{b}\right|\end{array}$$

We know that vectors a , b and their difference a - b are represented by three side of a triangle OAE. Further we know that a side of triangle is always less than the sum of remaining two sides. It means that sum of two sides is greater than the third side :

$$\begin{array}{c}\mathrm{OE}+\mathrm{AE}>\mathrm{OA}\\ \Rightarrow \mathrm{OE}>\mathrm{OA}-\mathrm{AE}\\ \Rightarrow |\mathbf{a}-\mathbf{b}|>\left|\mathbf{a}\right|-\left|\mathbf{b}\right|\end{array}$$

There is one possibility, however, that two vectors a and b are collinear and act in the opposite directions. In that case, magnitude of their difference will be equal to the difference of the magnitudes of individual vector. This magnitude represents the minimum or least magnitude of two vectors being combined.

$$\begin{array}{c}\Rightarrow \mathrm{OE}=\mathrm{OA}-\mathrm{AE}\\ \Rightarrow |\mathbf{a}-\mathbf{b}|=\left|\mathbf{a}\right|-\left|\mathbf{b}\right|\end{array}$$

Combining two results, we have :

$$\begin{array}{c}|\mathbf{a}-\mathbf{b}|\ge \left|\mathbf{a}\right|-\left|\mathbf{b}\right|\end{array}$$

Lami's theorem

Lami's theorem relates magnitude of three non-collinear vectors with the angles enclosed between pair of two vectors, provided resultant of three vectors is zero (null vector). This theorem is a manifestation of triangle law of addition. According to this theorem, if resultant of three vectors a , b and c is zero (null vector), then

$$\begin{array}{c}\frac{a}{\sin \alpha }=\frac{b}{\sin \beta }=\frac{c}{\sin \gamma }\end{array}$$

where α, β and γ be the angle between the remaining pairs of vectors.

We know that if the resultant of three vectors is zero, then they are represented by three sides of a triangle in magnitude and direction.

Considering the magnitude of vectors and applying sine law of triangle, we have :

$$\begin{array}{c}\frac{\mathrm{AB}}{\sin \mathrm{BCA}}=\frac{\mathrm{BC}}{\sin \mathrm{CAB}}=\frac{\mathrm{CA}}{\sin \mathrm{ABC}}\end{array}$$

$$\begin{array}{c}\Rightarrow \frac{\mathrm{AB}}{\sin (\pi -\alpha )}=\frac{\mathrm{BC}}{\sin (\pi -\beta )}=\frac{\mathrm{CA}}{\sin (\pi -\gamma )}\end{array}$$

$$\begin{array}{c}\Rightarrow \frac{\mathrm{AB}}{\sin \alpha }=\frac{\mathrm{BC}}{\sin \beta }=\frac{\mathrm{CA}}{\sin \gamma }\end{array}$$

It is important to note that the ratio involves exterior (outside) angles – not the interior angles of the triangle. Also, the angle associated with the magnitude of a vector in the individual ratio is the included angle between the remaining vectors.

Exercises