The components of vector 𝐀 are Ax and A, (both positive), and the angle that it makes with respe

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The components of vector 𝐀 are Ax and A, (both positive),...

The components of vector $\overrightarrow{\mathbf{A}}$ are $A_{x}$ and $A,$ (both positive), and the angle that it makes with respect to the positive $x$ axis is $\theta$. Find the angle $\theta$ if the components of the displacement vector $\overrightarrow{\mathbf{A}}$ are (a) $A_{x}=12 \mathrm{m}$ and $A_{y}=12 \mathrm{m},$ (b) $A_{x}=17 \mathrm{m}$ and $A_{y}=12 \mathrm{m},$ and (c) $A_{x}=12 \mathrm{m}$ and $A_{y}=17 \mathrm{m}$.

Key Concepts

Vector Components

Vectors in a plane can be decomposed into their components along the x and y axes, which represent the projections of the vector in each of these directions. This decomposition allows the vector to be studied using the principles of right triangle geometry and simplifies calculations involving direction and magnitude.

Cartesian Coordinate System

The Cartesian coordinate system provides a framework using perpendicular axes (typically labeled x and y) for representing points and vectors in a plane. It is essential for breaking down vectors into components and for defining angles with respect to a reference axis, such as the positive x-axis.

Right Triangle Trigonometry

Right triangle trigonometry involves the relationships between the angles and sides of a right triangle. In the context of vector decomposition, the x and y components form the adjacent and opposite sides of a right triangle, respectively. This relationship is used to determine the angle of the vector relative to the horizontal axis.

Inverse Tangent Function

The inverse tangent function, often denoted as arctan, is used to calculate the angle of a right triangle when the lengths of the opposite and adjacent sides are known. In vector analysis, the angle that a vector makes with the positive x-axis can be found by taking the arctan of the ratio of the y component to the x component.

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