Given |A1|=2,|A2|=3 and |A1+A2|=3. Find the value of (A1+2 A2) ·(3 A1-4 A2) (A) -64 (B) 60 (C) -

step 1 In this question we will learn about some basic concept of the vectors. The given question we ha

Given |A1|=2,|A2|=3 and |A1+A2|=3. Find the value of (A1+2...

Given $\left|\overrightarrow{A_{1}}\right|=2,\left|\overrightarrow{A_{2}}\right|=3$ and $\left|\overrightarrow{A_{1}}+\overrightarrow{A_{2}}\right|=3$. Find the value of $\left(\overrightarrow{A_{1}}+2 \overrightarrow{A_{2}}\right) \cdot\left(3 \overrightarrow{A_{1}}-4 \overrightarrow{A_{2}}\right)$ (A) $-64$ (B) 60 (C) $-60$ (D) 64

Given $\left|\overrightarrow{A_{1}}\right|=2,\left|\overrightarrow{A_{2}}\right|=3$ and $\left|\overrightarrow{A_{1}}+\overrightarrow{A_{2}}\right|=3$. Find the
value of $\left(\overrightarrow{A_{1}}+2 \overrightarrow{A_{2}}\right) \cdot\left(3 \overrightarrow{A_{1}}-4 \overrightarrow{A_{2}}\right)$
(A) $-64$
(B) 60
(C) $-60$
(D) 64

Key Concepts

Distributive Property of the Dot Product

The distributive property allows the dot product to be expanded over the addition of vectors, meaning that the dot product of a sum of vectors with another vector equals the sum of the individual dot products. This property is key to simplifying and evaluating expressions involving multiple vectors and their interactions.

Scalar Multiplication

Scalar multiplication is the process of multiplying a vector by a scalar, resulting in a new vector whose magnitude is scaled by the scalar and whose direction remains the same if the scalar is positive, or reverses if negative. This concept is important for adjusting the magnitude of vectors and for forming linear combinations.

Vector Addition

Vector addition involves combining two or more vectors to form a new vector. This operation is done component-wise and is crucial in problems that involve the summing of different vector quantities, often representing combined effects or forces.

Dot Product

The dot product is an algebraic operation that takes two equal-length vectors and returns a scalar. It encapsulates both the magnitude of the vectors and the cosine of the angle between them, and is essential for understanding projections, angles, and orthogonality in vector spaces.

Vector Magnitude

This concept refers to the length or size of a vector, typically computed using the square root of the sum of the squares of its components. It is fundamental in comparing vectors and understanding their geometric properties, such as when determining distances or norms in various spaces.

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