Given |A⃗1|=2,|A⃗2|=3 and |A⃗1+A⃗2|=3 . Find the value of (A⃗1+2 A⃗2) ·(3 A⃗1-4 A⃗2) a. 64 b. 60

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Given |A⃗1|=2,|A⃗2|=3 and |A⃗1+A⃗2|=3 . Find the value of...

Given $\left|\vec{A}_{1}\right|=2,\left|\vec{A}_{2}\right|=3$ and $\left|\vec{A}_{1}+\vec{A}_{2}\right|=3 .$ Find the
value of $\left(\vec{A}_{1}+2 \vec{A}_{2}\right) \cdot\left(3 \vec{A}_{1}-4 \vec{A}_{2}\right)$
a. 64
b. 60
c. 62
d. 61

Key Concepts

Dot Product

The dot product is an operation that takes two vectors and outputs a scalar, reflecting the degree to which the vectors point in the same direction. Its properties include commutativity and distributivity over addition, and it can also be expressed in terms of the magnitudes of the vectors and the cosine of the angle between them.

Vector Magnitude

The magnitude of a vector is a measure of its length or size and is calculated as the square root of the dot product of the vector with itself. It is essential for assessing the scale of vectors and is used in various computations, including those involving vector sums.

Distributive Property of the Dot Product

The dot product being distributive means that it obeys the rule that the dot product of a vector with a sum of two vectors is equal to the sum of the dot products of that vector with each of the individual vectors. This property is instrumental in expanding and simplifying expressions involving combinations of vectors.

Relationship Between Dot Product and Angle

The expression of the dot product in terms of the magnitudes of two vectors and the cosine of the angle between them bridges the algebraic and geometric interpretations of vectors. This relationship is vital for problems that involve finding unknown angles or relating vector magnitudes when combined vector expressions are involved.

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