Find the component form of v given its magnitude and the...

Find the component form of v given its magnitude and the angle it makes with the positive $x$ -axis. Sketch v. $$\begin{array}{cc}\text{Magnitude} && \text{Angle} \\ \|\mathbf{v}\|=1 &&
\theta=45^{\circ} \end{array}$$

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Key Concepts

Vector Component Form

The vector component form represents a vector as an ordered set of numbers, each corresponding to its projection along the coordinate axes. This form simplifies many mathematical operations such as addition, subtraction, and finding the dot product, by allowing vectors to be manipulated algebraically.

Magnitude and Direction

A vector is characterized by both a magnitude and a direction. The magnitude quantifies the length or size of the vector, while the direction, often specified as an angle relative to a reference axis, indicates its orientation in space. Together, they provide a complete description of the vector.

Trigonometric Decomposition

When a vector’s magnitude and the angle it makes with an axis are known, the vector can be decomposed into its horizontal and vertical components using trigonometric functions. The cosine of the angle gives the ratio for the component along the x-axis, while the sine of the angle gives the ratio for the component along the y-axis. This method of using sine and cosine is fundamental for converting between polar and rectangular forms of a vector.

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