A car is traveling at a speed of 50 km / hr. The positive y -axis is north and the positive x -a

step 1 Okay, our car is going to be speeding at 50 kilometers per hour. We're given different direction

A car is traveling at a speed of 50 km / hr. The positive y...

A car is traveling at a speed of $50 \mathrm{km} / \mathrm{hr}$. The positive $y$ -axis is north and the positive $x$ -axis is east. Resolve the car's velocity vector (in 2 -space) into components if the car is traveling in each of the following directions: (a) East (b) South (c) Southeast (d) Northwest.

Key Concepts

Unit Vectors

Unit vectors are vectors with a magnitude of one that indicate direction and are used to define the axes in a coordinate system. When decomposing a vector, these unit vectors serve as the basis for scaling the magnitude appropriately in each direction, ensuring that the original vector’s direction and magnitude are preserved in its component form.

Trigonometric Components

Trigonometric components involve using sine and cosine functions to resolve a vector into its horizontal and vertical parts based on its magnitude and the given angle relative to the coordinate axes. This technique is crucial when vectors are oriented at an angle, such as diagonal directions like southeast or northwest.

Coordinate System

A coordinate system, such as the Cartesian coordinate system, provides a framework for representing points and vectors in space using a set of numerical coordinates. In this context, defining the positive directions (north for the y-axis and east for the x-axis) allows clear interpretation of vector directions and facilitates the computation of components.

Vector Decomposition

Vector decomposition is the process of breaking a vector into its orthogonal components, typically along the axes of a coordinate system. This allows a vector to be expressed in terms of its effects in different directions, which is essential in physics and engineering for analyzing motion, forces, and other vector quantities.

Transcript

Please give Ace some feedback