Question

Two vector A and B are added. Prove that the magnitude of the resultant vector cannot be greater that (A+B) and smaller than (A-B) or (B-A)

Answer 1

Formula of resultant of vector,

$R= \sqrt{ {p+ {q}^{2} +2pq \times cos(α)}}$

cos(α) is maximum for 0 and minimum for 180

so,

for maximum α=0

$R = \sqrt{ {p}^{2} + {q}^{2} + 2pq \cos(0) }$

$R = \sqrt{ {p}^{2} + {q}^{2} + 2pq \times 1 }$

$R = \sqrt{ {p}^{2} + {q}^{2} + 2pq}$

$R = \sqrt{(p + q} )^{2}$

$R = p + q$

so,

magnitude of resultant vector cannot be more than (A+B)

for minimum α=180

$R = \sqrt{ {p}^{2} + {q}^{2} + 2pq \cos(180) }$

$R = \sqrt{ {p}^{2} + {q}^{2} + 2pq \cos(90 + 90) }$

$R = \sqrt{ {p}^{2} + {q}^{2} - 2pq \ \sin (90) }$

$R = \sqrt{ {p}^{2} + {q}^{2} - 2pq }$

$R = \sqrt{ ({p} - q) ^{2} }$

$R = p - q$

or

$R = q - p$

so,

magnitude of reusltant vector cannot be less than (A-B) or (B-A)