Write the vector 𝐯 in the form a 𝐢+b 𝐣, given its magnitude 𝐯 and the angle αit makes with the p

step 1 In this question, the model is of V is given as 15 units and this vector V, making an angle of a

Write the vector 𝐯 in the form a 𝐢+b 𝐣, given its magnitude...

Write the vector $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j},$ given its magnitude $\|\mathbf{v}\|$ and the angle $\alpha$ it makes with the positive $x$ -axis.
$$\|\mathbf{v}\|=15, \quad \alpha=315^{\circ}$$

Key Concepts

Angle Measurement and Quadrants

Angles in vector problems are typically measured from the positive x-axis, and the sign of the vector components depends on the quadrant in which the vector lies. Proper interpretation of the angle, including using known values and periodicity of trigonometric functions, ensures that the vector components are correctly determined based on the given directional information.

Vector Representation

This concept refers to expressing vectors in terms of their components along the principal axes of a coordinate system. In two dimensions, vectors are commonly written as a combination of horizontal and vertical components using the standard unit vectors, typically denoted i (for the x-axis) and j (for the y-axis). This format allows for easy manipulation in algebraic operations and facilitates the interpretation of directional quantities.

Polar Coordinates and Vector Decomposition

Vectors can also be described by their magnitude and the angle they make with the positive x-axis, which is essentially a polar coordinate description. Converting from polar to Cartesian coordinates involves decomposing the vector into its horizontal and vertical components. This is done using the relationships: horizontal component = magnitude × cos(angle) and vertical component = magnitude × sin(angle).

Trigonometric Functions

Trigonometric functions such as cosine and sine are fundamental in converting between polar and rectangular forms. They provide the means to calculate the lengths of the projections of a vector on the horizontal and vertical axes respectively. Understanding these functions is crucial when working with vectors defined by an angle and a magnitude.

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