Question

The maximum and minimum magnitude of the
resultant of two given vectors are 17 units and
7 unit respectively. If these two vectors are at right
angles to each other, then find the magnitude of
their resultant.

Answer 1

$\rule{200}{2}$

ANSWER:

• The magnitude of their resultant, when angle between them is 90° = 169

$\rule{200}{2}$

GIVEN:

• Maximum magnitude of the resultant of two given vectors is 17 units

• Minimum magnitude of the resultant of two given vectors is 7 units

$\rule{200}{2}$

TO FIND:

• The magnitude of their resultant, when angle between them is 90°

$\rule{200}{2}$

EXPLANATION:

$Let \: \vec{A} \: \: and \: \: \vec{B} \: \: be \: \: the \: \: two \: \: vectors.$

$R= \sqrt{A^2 + B^2 + 2AB \cos \theta}$

For maximum magnitude θ = 0°

$17 = \sqrt{A^2 + B^2 + 2AB \cos {0}^{\circ} } \\ \\ \\ 17 = \sqrt{A^2 + B^2 + 2AB } \\ \\ \\ 17 = \sqrt{(A+ B)^2} \\ \\ \\ 17 = A+ B$

For minimum magnitude θ = 180°

$7 = \sqrt{A^2 + B^2 + 2AB \cos {180}^{\circ} } \\ \\ \\ 7 = \sqrt{A^2 + B^2 - 2AB } \\ \\ \\ 7 = \sqrt{(A - B)^2} \\ \\ \\ 7 = A - B$

We got A + B = 17 and A - B = 7

Add those two terms

A + B + A - B = 17 + 7

2A = 24

A = 12

Substitute A = 12 in A - B = 7

12 - B = 7

B = 12 - 7

B = 5

Now we want to find the resultant when θ = 90°

$R= \sqrt{A^2 + B^2 + 2AB \cos {90}^{\circ} } \\ \\ \\ R = \sqrt{12^2 + 5^2 + 0 } \\ \\ \\R = 144 + 25 \\ \\ \\ R= 169$

Hence the magnitude of their resultant, when θ = 90° is 169.

$\rule{200}{2}$