When Studying Vectors, How Would I Go About Answering This Question?
In Each of the following Cases, find the magnitude and direction of the resultant vectors of the two given vectors:
a) a displacement of magnitude 26km on a bearing of 175 degrees and a displacement of magnitude 18km on a bearing 294 degrees
b) Forces of 355 N on a bearing of 320 degrees and 270 N on a bearing of 025 degrees
help would be MUCH appreciated, if you could also explain how you answer it ?
also if you feel like drawing a diagram on paint or something if you could link it?
THANKS :D
To answer this question, you will need to understand vector addition and how to break down vectors into their components. Here's how you can approach it:
a) For vector addition, you can use the method of components. Start by drawing a diagram of the two displacements.
1. Draw a vector representing the displacement of 26 km on a bearing of 175 degrees. The magnitude of this vector is 26 km, and the direction is 175 degrees counterclockwise from the positive x-axis.
2. From the end point of the first vector, draw another vector representing the displacement of 18 km on a bearing of 294 degrees. The magnitude of this vector is 18 km, and the direction is 294 degrees counterclockwise from the positive x-axis.
To find the resultant vector, you need to add the x-components and y-components of the two vectors individually.
3. Break down the two vectors into their x and y components. To do this, you can use trigonometry. Let's assume the positive x-axis is in the east direction, and the positive y-axis is in the north direction.
The first vector has an angle of 175 degrees counterclockwise from the positive x-axis. To find its x-component, you can use the formula: x-component = magnitude * cos(angle). Similarly, the y-component can be found using the formula: y-component = magnitude * sin(angle).
For the first vector:
x-component = 26 km * cos(175 degrees)
y-component = 26 km * sin(175 degrees)
Similarly, for the second vector:
x-component = 18 km * cos(294 degrees)
y-component = 18 km * sin(294 degrees)
4. Add up the x-components and y-components separately. The resultant vector's x-component is the sum of the x-components of the two vectors, and the resultant vector's y-component is the sum of the y-components of the two vectors.
Resultant x-component = x-component of the first vector + x-component of the second vector
Resultant y-component = y-component of the first vector + y-component of the second vector
5. To find the magnitude of the resultant vector, you can use the Pythagorean theorem: magnitude = sqrt(x^2 + y^2), where x is the resultant x-component and y is the resultant y-component.
6. To find the direction of the resultant vector, you can use the inverse tangent function: direction = atan(y/x). Make sure to consider the appropriate quadrant for the direction.
You can use this step-by-step explanation to solve part a) of the question. Similarly, you can apply the same approach to solve part b) involving forces.
Unfortunately, as a text-based AI, I cannot provide links to diagrams. However, you can manually draw a diagram on paper or use online tools like Paint or drawing apps to visualize the vectors.
Hope this helps! Let me know if you have any further questions.
in order to add vectors, it's necessary to convert to (x,y) from (r,θ)
For problem (a), we want to add
(26,175°) and (18,294°)
Now, bearings use 0° as North, measured clockwise, so we have to use
x = r*sinθ
y = r*cosθ
(26,175°) -> (2.266,-25.901)
(18,294°) -> (-16.444,7.321)
add 'em up to get
(-14.178,-18.580)
convert back to (r,θ) to get distance and bearing.
now go and do (b) the same way