The resultant of two vector A and B is perpendicular to vector A and its magnitude is equal to half of magnitude of vector B then the angle between A and B?

Question

Answer

Solution

Let Vector A and Vector B be represented as A and B respectively.

Let the angle between Vector A and Vector B be represented as θ.

Let the magnitude of Vector B be represented as |B|.

Given that the resultant of Vector A and Vector B is perpendicular to Vector A and its magnitude is equal to half of the magnitude of Vector B.

This can be represented as A + B = C where C is perpendicular to A and |C| = 0.5|B|.

The dot product of two perpendicular vectors is zero. Therefore, A.C = 0.

Using this, we can write A.(A+B) = 0. This simplifies to A.A + A.B = 0.

Since A.A = |A|^2 and |A| is always positive, we can divide both sides by |A|^2 to get 1 + (A.B)/|A|^2 = 0.

Substituting cos(θ) = (A.B)/(|A||B|), we get cos(θ) = -1/2.

Therefore, θ = 120 degrees or 2π/3 radians.

Explanation

The problem involves finding the angle between two vectors A and B given that the resultant of the two vectors is perpendicular to A and has half the magnitude of B. To solve the problem, we can use the concept of dot product between two vectors and the fact that the resultant of two vectors is obtained by adding them together.

Representing the Vectors

Let Vector A and Vector B be represented as A and B respectively. Let the angle between Vector A and Vector B be represented as θ. Let the magnitude of Vector B be represented as |B|.

Using Resultant Vector

Given that the resultant of Vector A and Vector B is perpendicular to Vector A and its magnitude is equal to half of the magnitude of Vector B. This can be represented as A + B = C where C is perpendicular to A and |C| = 0.5|B|.

Using Dot Product

The dot product of two perpendicular vectors is zero. Therefore, A.C = 0. Using this, we can write A.(A+B) = 0. This simplifies to A.A + A.B = 0.

Substituting and Solving

Since A.A = |A|^2 and |A| is always positive, we can divide both sides by |A|^2 to get 1 + (A.B)/|A|^2 = 0. Substituting cos(θ) = (A.B)/(|A||B|), we get cos(θ) = -1/2. Therefore, θ = 120 degrees or 2π/3 radians.

Answer

150

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