Question

Two vectors acting in opposite directions have a resultant of 10 units. If they are at right angles to each other then the resultant is 50 units. Calculate the magnitude of two vectors .​

Answer 1

Answer:

Magnitude of the two vectors = 40 units and 30 units

Explanation:

Given that,

Two vectors acting in opposite directions

so,

angle between the two vectors = 180°

given resultant in this condition = 10 units.

let the two vectors a^-> and b^->

a^-> + b^-> = a² + b² + 2abcosθ

a² + b² + 2abcos 180° =10²

a² + b² + 2ab(-1) = 10²

a² + b² - 2ab = 10²

(a - b)² = 10²

a - b = 10 ...(1)

now,

also given that,

the two vectors are at right angles to each other

now in this condition

θ = 90°

and given resultant in this condition = 50 units,

50² = a² + b² + 2abcos 90°

a² + b² + 2ab(0) = 50²

a² + b² = 2500

a² + b² - 2ab + 2ab = 2500

(a - b)² + 2ab = 2500

from (1)

a - b = 10

10² + 2ab = 2500

100 + 2ab = 2500

2ab = 2500 - 100

2ab = 2400

ab = 2400/2

ab = 1200 ....(2)

from equation (1)

a - b = 10

a = b + 10 ...(3)

putting the value of a in (2)

ab = 1200

(b + 10)b = 1200

b² + 10b - 1200 = 0

b² - 30b + 40b - 1200 = 0

b(b - 30) + 40(b - 30) = 0

(b - 30) (b + 40)

b = 30, -40

since,

magnitude can't be negative

so,

b = 30

now,

putting the value of b on (3)

a = b + 10

a = 30 + 10

a = 40

b = 30

so,

Magnitude of the two vectors

= 40 units and 30 units

Answer 2

» Two vectors acting in opposite directions have a resultant of 10 units.

Means the angle between the two vectors is 180°.

• Let one vector be A and another vector be B.

According to triangles law..

=> A² + B² + 2AB cos Ø = (10)²

Here Ø = 180°

=> A² + B² + 2AB cos 180° = 100

We know that cos 180° = (-1)

=> A² + B² + 2AB (-1) = 100

=> A² + B² - 2AB = 100

(A - B)² = A² + B² - 2AB

=> (A - B)² = 100

=> A - B = √100

=> A - B = 10 _______ (eq 1)

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» If they are at right angles to each other then the resultant is 50 units.

Now the vectors are right angles to each other. Means the angle between them is 90°.

=> A² + B² + 2AB cos 90° = (50)²

We know that cos 90° = 0

=> A² + B² + 2AB (0) = 2500

=> A² + B² = 2500

=> A² + B² - 2AB + 2AB = 2500

(A - B)² = A² + B² - 2AB

=> (A - B)² + 2AB = 2500

=> (10)² + 2AB = 2500

=> 100 + 2AB = 2500

=> 2AB = 2400

=> AB = 1200

=> A = $\dfrac{1200}{B}$ _____ (eq 2)

• Put value of B in (eq 1)

=> $\dfrac{1200}{B}$ - B = 10

=> $\dfrac{1200 \: - \: {B}^{2} }{B}$ = 10

=> 1200 - B² = 10B

=> B² + 10B - 1200 = 0

=> B² + 40B - 30B - 1200 = 0

=> B(B + 40) - 30(N + 40) = 0

=> (B - 30) (B + 40) = 0

• B - 30 = 0

=> B = 30

• B + 40 = 0

=> B = - 40 (neglected)

Put value of B in (eq 2)

=> A = $\dfrac{1200}{30}$

=> A = 40

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• We have to fimd the magnitude kd the two vectors. (Means A and B)

From above calculations we have A = 40 and B = 30

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$\textbf{Magnitude of two vectors is 40 and 30}$

$\textbf{units.}$

__________ [ANSWER]

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