Given the displacement vectors 𝐀=(3 𝐢̂-4 𝐣̂+4 𝐤̂) m and 𝐁=(2...

Given the displacement vectors $\overrightarrow{\mathbf{A}}=(3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \mathrm{m}$ and $\overrightarrow{\mathbf{B}}=(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-7 \hat{\mathbf{k}}) \quad \mathrm{m}, \quad$ find $\quad$ the $\quad$ magnitudes $\quad$ of the following vectors and express each in terms of its rectangular components. (a) $\overrightarrow{\mathbf{C}}=\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ (b) $\overrightarrow{\mathbf{D}}=$ $2 \overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$.

Given the displacement vectors $\overrightarrow{\mathbf{A}}=(3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \mathrm{m}$
and $\overrightarrow{\mathbf{B}}=(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-7 \hat{\mathbf{k}}) \quad \mathrm{m}, \quad$ find $\quad$ the $\quad$ magnitudes $\quad$ of the following vectors and express each in terms of its rectangular components. (a) $\overrightarrow{\mathbf{C}}=\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ (b) $\overrightarrow{\mathbf{D}}=$ $2 \overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$.

Key Concepts

Calculation of Vector Magnitude

The magnitude of a vector, often calculated using the Euclidean norm, is the length or size of the vector. It is determined by taking the square root of the sum of the squares of its components, providing a measure of how long the vector is in the vector space.

Vector Addition

Vector addition is the process of combining two or more vectors to produce a resultant vector by adding their corresponding components. This operation respects the dimensions of the space and yields a new vector whose direction and magnitude depend on the individual vectors being added.

Vector Subtraction

Vector subtraction involves determining the difference between two vectors by subtracting the corresponding components of one vector from the other. It is essentially adding a negative vector, which changes the direction of the subtracted vector, resulting in a new vector that represents the displacement from one vector to the other.

Scalar Multiplication

Scalar multiplication is an operation where a vector is multiplied by a scalar (a real number), scaling the magnitude of the vector while preserving or reversing its direction depending on the sign of the scalar. This operation is applied to each component of the vector individually.

Rectangular Components Representation

Rectangular components representation expresses a vector in a coordinate system, typically in terms of unit vectors along the x, y, and z axes. This method breaks a vector down into parts that represent its projection along each axis and facilitates straightforward vector operations such as addition, subtraction, and scaling.

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