If |A⃗ ×B⃗|=√(3) A⃗ ·B⃗, then the value of |A⃗+B⃗| is (A) (A^2+B^2+A B)^1 / 2 (B) (A^2+B^2+(A B)

step 1 In this question we will learn about some basic concept of the vectors. The question is, if the

If |A⃗ ×B⃗|=√(3) A⃗ ·B⃗, then the value of |A⃗+B⃗| is (A)...

If $|\vec{A} \times \vec{B}|=\sqrt{3} \vec{A} \cdot \vec{B}$, then the value of $|\vec{A}+\vec{B}|$ is
(A) $\left(A^{2}+B^{2}+A B\right)^{1 / 2}$
(B) $\left(A^{2}+B^{2}+\frac{A B}{\sqrt{3}}\right)^{1 / 2}$
(C) $(A+B)$
(D) $\left(A^{2}+B^{2}+\sqrt{3} A B\right)^{1 / 2}$

Key Concepts

Vector Cross Product

The cross product of two vectors is an operation that produces a vector orthogonal to both original vectors. Its magnitude is given by the product of the magnitudes of the two vectors and the sine of the angle between them. This concept is central to problems that involve determining the perpendicular component or rotational effects in a vector space.

Vector Dot Product

The dot product is an operation that takes two vectors and returns a scalar quantity equal to the product of their magnitudes and the cosine of the angle between them. It is widely used to quantify how much one vector extends in the direction of another and plays a crucial role in establishing angular relationships between vectors.

Angle Between Vectors

Determining the angle between two vectors involves both the cross and dot products via the sine and cosine of the angle, respectively. By comparing these quantities, one can derive conditions or constraints on the angle, which are essential for solving vector equations without directly being given the angle.

Magnitude of Vector Sum

The magnitude of the sum of two vectors is computed using the law of cosines for vectors. It expresses the squared magnitude as the sum of the squares of the individual magnitudes plus twice their product times the cosine of the angle between them. This concept is fundamental in vector addition and is applicable in a variety of contexts, from physics to engineering.

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