Assume that 𝐯=2,w=3, and the angle between 𝐯 and w is 120^∘ . Determine: (a) 𝐯·w (b) 2

step 1 We know that v .W is equal to the magnitude of v times the magnitude of w times the cosine of th

Assume that 𝐯=2,w=3, and the angle between 𝐯 and w is 120^∘...

Assume that $\|\mathbf{v}\|=2,\|\mathrm{w}\|=3,$ and the angle between $\mathbf{v}$ and $\mathrm{w}$ is $120^{\circ} .$ Determine: \begin{equation}\begin{array}{ll}{\text { (a) } \mathbf{v} \cdot \mathrm{w}} \quad\quad {\text { (b) }\|2 \mathbf{v}+\mathrm{w}\|} \quad\quad {\text { (c) }\|2 \mathbf{v}-3 \mathrm{w}\|}\end{array}\end{equation}

Assume that $\|\mathbf{v}\|=2,\|\mathrm{w}\|=3,$ and the angle between $\mathbf{v}$ and $\mathrm{w}$ is $120^{\circ} .$ Determine:
\begin{equation}\begin{array}{ll}{\text { (a) } \mathbf{v} \cdot \mathrm{w}} \quad\quad {\text { (b) }\|2 \mathbf{v}+\mathrm{w}\|} \quad\quad {\text { (c) }\|2 \mathbf{v}-3 \mathrm{w}\|}\end{array}\end{equation}

Key Concepts

Magnitude of a Linear Combination

Calculating the magnitude of a linear combination of vectors involves combining the effects of scalar multiplication and vector addition. This process often utilizes the distributive property of the dot product, where the square of the norm of the resulting vector is determined by summing the squares of the scaled norms of the individual vectors along with twice the product of the scalars and the dot product of the vectors. This concept is critical in many applications, including optimization and solving systems in linear algebra.

Scalar Multiplication and Vector Addition

Scalar multiplication and vector addition are basic operations in vector spaces that allow for the construction of new vectors from existing ones. Scalar multiplication scales a vector by a constant, while vector addition combines two vectors to produce a resultant vector. These operations form the basis of forming linear combinations, which are common in various applications such as physics and engineering.

Angle Between Vectors

The angle between vectors is a measure of the directional difference between them. In the context of the dot product, the cosine of this angle is used to bridge the gap between the algebraic and geometric representations of vectors. This concept is vital for determining the degree of alignment or orthogonality of vectors.

Vector Norm

The vector norm, often referred to as the magnitude or length of a vector, quantifies the size of the vector in a Euclidean space. It is typically calculated as the square root of the sum of the squares of its components. Understanding vector norms is key to comparing vectors and is critical in normalization processes where vectors are scaled to have unit length.

Dot Product

The dot product is a fundamental operation in vector algebra that combines two vectors to produce a scalar. It is defined as the product of the magnitudes of the two vectors and the cosine of the angle between them. This operation is essential for determining how much one vector extends in the direction of another, and it plays a crucial role in calculations involving projections and angles between vectors.

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