Components of a Vector Find the horizontal and vertical...

Components of a Vector Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors $\mathbf{i}$ and $\mathbf{j}$.
$|\mathbf{v}|=50, \quad \theta=120^{\circ}$

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Key Concepts

Component Form Representation

Writing a vector in component form means expressing it as a sum of its horizontal and vertical components multiplied by the unit vectors, typically denoted as i (for the horizontal axis) and j (for the vertical axis). This form is useful for vector addition, subtraction, and other algebraic operations.

Standard Basis Vectors

Standard basis vectors, usually represented as i and j in two-dimensional space, define the directions of the axes. They serve as the building blocks for any vector in the plane, allowing the vector to be expressed as a linear combination of these basis vectors.

Vector Decomposition

This concept involves breaking down a vector into its orthogonal components (typically horizontal and vertical) so that the vector can be analyzed or reconstructed from these simpler parts. Decomposing a vector simplifies many calculations in physics and engineering, especially when combining multiple vectors.

Trigonometric Functions

Trigonometric functions, particularly cosine and sine, are used to calculate the horizontal and vertical components of a vector from its magnitude and direction. The cosine function is used for the horizontal component and the sine function for the vertical component when the angle is measured from the positive x-axis.

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