91Ó°ÊÓ

Vector summation, or vector addition, is an operation performed to determine the resultant vector of two or more vectors.

The magnitude of the first vector,$V_1=3.5\mathrm{km}$

The magnitude of the second vector,$V_2=4.0\mathrm{km}$

Let us assume that the angle between the first and second vectors is $\mathrm{θ}$and the magnitude of the resultant vector is V.

When two vectors $\stackrel{→}{A}$and $\stackrel{→}{B}$ having magnitude A and B, respectively, are aligned at an angle $\mathrm{θ}$with respect to each other, the magnitude of the vector sum of these two vectors is written as$R=\sqrt{A^2+B^2+2AB\cos \mathrm{θ}}$

Here, R is the magnitude of the resultant vector.

The length of a vector signifies the magnitude of that vector. Thus, using the formula mentioned above, the magnitude of the resultant vector can be written as$V=\sqrt{V_1^2+V_2^2+2V_1{V}_2\cos \mathrm{θ}}$

From the above equation of V, you can see that the value of V is dependent on $\cos \mathrm{θ}$. When $\cos \mathrm{θ}$ will be the maximum, V will also be the maximum. $\cos \mathrm{θ}$is the maximum at $\mathrm{θ}=0°$. ($\cos 0°=1$)

Thus, the maximum value of the vector sum will be

$V=\sqrt{V_1^2+V_2^2+2V_1{V}_2\cos 0°}$

Substituting the values in the above equation, you get:

$\begin{array}{c}{V}_{\max }=\sqrt{{3.5}^2+{4.0}^2+\left(2×3.5×4.0\right)}\\ =7.5\mathrm{km}\end{array}$

Thus, the maximum value of the vector sum will be 7.5 km.

From the equation of V, you can see that when $\cos \mathrm{θ}$will be the minimum, V will also be the minimum. $\cos \mathrm{θ}$is the minimum at $\mathrm{θ}=180°$. ($\cos 180°=-1$)

Thus, the minimum value of the vector sum will be

$V=\sqrt{V_1^2+V_2^2+2V_1{V}_2\cos 180°}$

Substituting the values in the above equation, you get:

$\begin{array}{c}{V}_{\min }=\sqrt{{3.5}^2+{4.0}^2-\left(2×3.5×4.0\right)}\\ =0.5\mathrm{km}\end{array}$

Thus, the minimum value of the vector sum will be 0.5 km.