Solving vectors using component method

In summary, to add the given vectors using the component method, you must find the x and y components for each vector using the formula rcos(\theta) and rsin(\theta), where r is the magnitude of the vector and \theta is the direction in degrees. Once you have the components, you can add them for each direction to find the resultant vector. To find the magnitude of the resultant vector, use the Pythagorean theorem (a^2 + b^2 = c^2) where a and b are the x and y components, and c is the magnitude of the resultant vector. Finally, to find the direction of the resultant vector, use the inverse tangent function (tan^-1) to find the angle \theta
  • #1
yety124
2
0

Homework Statement


Add the following vectors using component method:
Vector A=175km; 30 degrees north of east
Vector B=153km; 20 degrees west of north
Vector C=195km; West
a)Find x and y component of each vector
b)Solve for the magnitude of resultant vector
c) Solve for direction(angle)

2. The attempt at a solution
So i these are what i got
Vector A: x=175cos 30...x=151.56 y=175sin 30...y=87.5
Vector B: x=153cos 20...x=-143.77 y=153sin 20...y=52.33
(are these correct?)
so after getting this i added all the x's and y's but then i realized that the question had said that vector C goes 195km west, so i now got confused what to do next.

please help thank you for your time.
 
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  • #2
There is nothing special about "in a westerly direction". A westerly vector contributes a component D.cos 180o in the x-direction, and D.sin 0o in the y-direction.
 
  • #3
well that was stupid of me, thanks for the reply finally solved it :D
 
  • #4
If [itex]\theta[/itex] is measured counterclockwise from the positive x-axis then a vector of length r and angle [itex]\theta[/itex] has components [itex]rcos(\theta)[/itex] and [itex]r sin(\theta)[/itex]. Strictly speaking, you are free to choose the "positive x-axis" any way you want as long as you are consistent but the usual convention is that the positive x-axis points East.

For the first vector you are given that r= 175km and the directon is 30 degrees north of east. "north of east" is counterclockwise from east so the angle is [itex]\theta= 30[/itex] degrees.

For the second vector you are given that r= 153km and the direction is 20 degrees west of north. West is clockwise of north but north itself is 90 degrees clockwise of east. The angle is [itex]\theta= 90+20= 110[/itex] degrees.

For the third vector you are given that r= 195km and the direction is west. West is exactly opposite east so the angle is [itex]\theta= 180[/itex] degrees.
 
  • #5



Your calculations for the x and y components of vector A and B are correct. As for vector C, since it only has a magnitude and direction (west), we can assume that the x component is -195 and the y component is 0. This is because going west means moving in the negative x direction, and there is no movement in the y direction.

To find the resultant vector, we can add all the x and y components together:
Rx = 151.56 + (-143.77) + (-195) = -187.21
Ry = 87.5 + 52.33 + 0 = 139.83

To find the magnitude of the resultant vector, we can use the Pythagorean theorem:
R = √(Rx^2 + Ry^2) = √((-187.21)^2 + (139.83)^2) = 234.51 km

To find the direction (angle) of the resultant vector, we can use inverse tangent:
θ = tan^-1(Ry/Rx) = tan^-1(139.83/-187.21) = -36.55 degrees

Therefore, the resultant vector has a magnitude of 234.51 km and is 36.55 degrees below the negative x-axis (west).

I hope this helps!
 

What is the component method for solving vectors?

The component method is a technique used to solve vector problems by breaking down a single vector into its horizontal and vertical components. This allows for easier computation and visualization of vector operations.

How do I find the horizontal and vertical components of a vector?

To find the horizontal and vertical components of a vector, you can use trigonometric functions such as sine, cosine, and tangent. The horizontal component can be found by multiplying the magnitude of the vector by the cosine of the angle it makes with the horizontal axis. The vertical component can be found by multiplying the magnitude of the vector by the sine of the angle.

Can the component method be used with vectors in any direction?

Yes, the component method can be used with vectors in any direction. The key is to choose appropriate axes and use the trigonometric functions to find the components.

How do I add or subtract vectors using the component method?

To add or subtract vectors using the component method, you can first find the horizontal and vertical components of each vector. Then, you can add or subtract the components separately to get the components of the resultant vector. Finally, you can use the Pythagorean theorem and trigonometric functions to find the magnitude and direction of the resultant vector.

What are some common applications of the component method?

The component method is commonly used in physics and engineering to solve problems involving forces and motion. It can also be used in navigation and surveying to calculate distances and directions. Additionally, the component method is useful in computer graphics and animation to represent and manipulate 2D and 3D vectors.

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