Question

two vectors having magnitude 10 units and 6 units act at a point at a point at a certain angle. the magnitude of the resultant of the two vectors is 18 units. is it true or false​

Answer 1

Given :-

◉ Two vectors A and B having magnitude 10 units and 6 units act at a point at a certain angle.

To Show :-

◉ The magnitude of the resultant vector of the two vectors A and B is 18 units.

Solution :-

For the resultant of two vectors, We use the following formula :-

⇒ R² = A² + B² + 2ABcos θ

Putting values, we get

⇒ (18)² = (10)² + (6)² + 2×10×6×cos θ

⇒ 324 = 100 + 36 = 120cosθ

⇒ 324 - 136 = 120cosθ

⇒ cos θ = 188/120

⇒ cos θ = 1.567

Which is impossible because range of cosine function lies within -1 and 1 .

So, It is not possible and hence False.

We can also solve by using a simple statement,

We know the maximum magnitude of resultant of two vectors is when the two vectors are added directly so they are at angle 0°.

But, In the question, It is given that they are at certain angle, So let the angle be dθ

⇒ A + B ≈ √(A² + B² + 2ABcos dθ)

dθ is infinitely small so we take a angle which is close to it i.e., 0° but it is not exactly 0 that's why we have a approximately sign.

⇒ A + B ≈ √(A + B)² [ a² + b² + 2ab = (a + b)²]

⇒ A + B ≈ |A| + |B|

⇒ A + B ≈ 10 + 6

⇒ A + B ≈ 16

Which is far less than 18 at this precision. Hence, It is not possible.