Compute the x -and y -components of the vectors A⃗, B⃗, C⃗ and D⃗ in Fig. 1.34

Name: Problem 35, Chapter 1: University Physics with Modern Physics
Uploaded: 2023-08-31T03:12:21-08:00
Duration: 4 min 2 s
Channel: Supratim Pal
Description: Compute the x -and y -components of the vectors A⃗, B⃗, C⃗ and D⃗ in Fig. 1.34

Compute the x -and y -components of the vectors A⃗, B⃗, C⃗...

Key Concepts

Angle Measurement and Conversion

Understanding how angles are measured and converted is crucial for accurately decomposing vectors. This involves identifying the correct reference angle, properly handling units (degrees or radians), and applying the appropriate trigonometric functions to find the vector components.

Cartesian Coordinate System

The Cartesian coordinate system provides a framework for expressing positions or vectors in a plane using perpendicular axes (usually labeled x and y). This system allows vectors to be defined by their horizontal and vertical displacements, which is essential for breaking them into components.

Vector Decomposition

Vector decomposition involves breaking a vector into its orthogonal components, typically along the x- and y-axes. By resolving a vector into these components, one can analyze its effect in each independent direction, simplifying calculations related to dynamics, statics, and other fields.

Trigonometric Functions

Trigonometric functions such as sine and cosine are fundamental tools in vector decomposition. They are used to calculate the magnitude of a vector's components based on its angle relative to the coordinate axes, where the x-component is obtained using the cosine function and the y-component using the sine function.

Transcript

00:01 Now here you have to compute the x and y components of the vectors a, b, c and d.

00:06 Right? so let me just draw the vectors that are given in this x, y coordinate system.

00:14 This is x, this is y.

00:15 Right here you have got vector a that is directly pointing downwards.

00:24 This is having magnitude of 8 meters.

00:27 Right? this is a.

00:29 Next you have got vector b which is the first quadrant.

00:34 This is b making an angle 30 degrees with the vertical and this magnitude is 15 meters.

00:44 Whereas you have got c that is sort of like this making an angle 25 degrees.

00:54 This is c vector.

00:56 This is 12 meters.

00:57 Right? in the third quadrant and there is d which is 53 degrees.

01:05 Right? and its magnitude is 10 meters.

01:10 So it's pretty simple.

01:12 So what you can do is you can have a vector that is having only the y component.

01:19 Right? so it's negative 8 j cap.

01:22 Right? so it has only y component.

01:25 I'm writing x components in i cap and the y components in terms of j cap.

01:32 Right? so it recommends it will be in meters.

01:35 So now vector b.

01:38 Vector b is making an angle 30 degrees with the vertical.

01:41 Right? so what we have is you can write 15 the magnitude times sine of 30 i cap.

01:53 Right? now the vector which is which is if you see here the component of the vector away from the angle is the sine component...

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