Question

If vector 1 have magnitude 5 unit.resultant of vector 1 and vector 2 have magnitude 15. which of the following value is a possible value of magnitude of vector 2
A) 9 B) 8 C) 7 D) 15​

Answer 1

If vector 1 has magnitude 5 units. resultant of vector 1 and vector 2 has magnitude 15 units.

which one of of the following values is a possible value of magnitude of vector 2.

A. 9   B. 8   C. 7    D. 15  

    let the vector 1 is a and the vector 2 is b.

here, magnitude of vector 1 = |a| = 5

magnitude of vector resultant of vector a and b , R =  15

we know, magnitude of resultant of two vector is given by, $R=\sqrt{|a|^2+|b|^2+2|a||b|cos\theta}$

⇒ 15 = $\sqrt{5^2+|b|^2+2\times5|b|cos\theta}$

⇒ 15² = 5² + |b|² + 10|b|cosθ

⇒ 225 - 25 = |b|² + 10|b|cosθ

⇒ |b|² + 10|b|cosθ - 200 = 0

we know the maximum value of cosθ can be 1 and minimum value of it can be -1

if cosθ = 1

|b|² + 10|b| - 200 = 0

⇒(|b| + 20)(|b| - 10) = 0

⇒|b| = 10, |b| ≠ -20 [ because |b| is absolute value of b]

if cosθ = -1

|b|² - 10|b| - 200 = 0

⇒(|b| + 10)(|b| - 20) = 0

⇒|b| = 20, |b| ≠ -10

hence, the value of |b| lies between 10 to 20.

I.e., 10 ≤ |b| ≤ 20

only option (D) 15 , has lies between [10, 20]

hence option (D) is correct choice.

Answer 2

If vector 1 have magnitude 5 unit. Resultant of vector 1 and vector 2 have magnitude 15 then possible value of magnitude of vector 2 is 15 units

Given:

Vector 1 have magnitude 5 unit.

Resultant of vector 1 and vector 2 have magnitude 15

To Find:

Which of the following value is a possible value of magnitude of vector 2

A) 9

B) 8

C) 7

D) 15​

Solution:

Resultant magnitude of Two Vectors with magnitudes A and B  lies between | A - B |  and | A + B |

Step 1:

Assume that vector 2 has magnitude  X

Step 2:

Resultant magnitude will lie in range

| X - 5 |  and | X + 5|

Step 3:

Resultant magnitude is 15 should lie between these 2 intervals

| X - 5 |  ≤  15 ≤  | X + 5|

10 ≤ X ≤ 20

Step 4:

Check the given options

only option D) 15 lies between 10 and 20

Hence correct option is  D) 15