A vector is given by 𝐑=2 𝐢̂+𝐣̂+3 𝐤̂ . Find (a) the...

A vector is given by $\overrightarrow{\mathbf{R}}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}} .$ Find (a) the magnitudes of the $x, y,$ and $z$ components; (b) the magnitude of $\overrightarrow{\mathbf{R}} ;$ and $(c)$ the angles between $\overrightarrow{\mathbf{R}}$ and the $x, y,$ and $z$ axes.

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Key Concepts

Direction Cosines

Direction cosines are the cosines of the angles between a vector and the coordinate axes. They relate the components of the vector to its magnitude and provide insight into the vector's orientation in space. The computation involves dividing each component by the vector's magnitude, establishing a direct link between geometry and vector algebra.

Vector Components

In the context of Cartesian coordinates, a vector can be decomposed into its components along the x, y, and z axes. These components represent the projection of the vector in each coordinate direction, and they form the basis for further calculations such as magnitude and directions. Understanding vector components is crucial for analyzing how a vector acts in multi-dimensional space.

Vector Magnitude

The magnitude of a vector is a measure of its length and is calculated by taking the square root of the sum of the squares of its components. This calculation is derived from the Pythagorean theorem, and it provides a scalar value that represents the size or extent of the vector irrespective of its direction.

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