How do I do this: add the following vectors , 7m/s [N30E] and 2m/s [S17E] I know that you must use trigonometry and break stuff up into components and use sine and cosine law; just not sure how. Thanks!
Do you just find the resultant?
Not sure how to go about adding up the angles. Thanks.
Yes, you find the resultant.
You don't add the angles. You add the components.
Letting north be the +y axis and east be the +x axis, the components of 7 m/s [N30E] are 7 sin 30 = 3.5 along the +x axis and 7 cos 30 = 6.062 along the +y axis. Calculate the components of the other vector similarly, and add the x and y components separately for the resultant.
I need to use trigonometry to do this though, not component law, is there any real difference?
If you want the magnitude of the resultant, use the law of cosines. Drawing a figure will help.
The law of sines can get you the sine of any angle of the triangle formed by the two velocity vectors and the resultant.
it's easier using components, but the answer will be the same.
Yes, ok I have done that, I drew a diagram, but the thing is, you have all these angles, the co interior angles, and I am unsure of which ones to use - you have the 30 from the NE and then there is 17 from SE but there is another one, and I am not sure how to find it. I know the equation would be R^2= (7)^2 (2)^2-2(7)(2)COS___ I am unsure about what the angle is. Thanks, sorry for being so confusing.
I could write down a bunch of equations but since I cannot draw the triangle for you with the tools we have, it would probably be difficult to explain in words what is going on.
Draw the two velocity vectors end to end. Then close the triangle to get the resultant. The angle between the two vectors that you were provided is 47 degrees. That is the angle that you use in the law of cosines to get the magnitude of the resultant
R^2 = 7^2 + 2^2 - 2*14 cos 47 = 33.90
R = 5.82 m/s
Now use the law of sines to get the other two angles of the triangle, which will tell you the direction of the resultant.
It is a lot easier adding components.
Thanks! I agree about the components but the question said specifically to use cosine and sine law :(. Thanks very much though!
I used the sine law to get 15 degrees. I am unsure as to what I now do with this angle. Do I add or subtract it somewhere to find the overall angle I am looking for? Thanks again! I owe you!
To add the given vectors, you need to break them down into their x and y components and then add the respective components together. Here's how you can solve it step by step:
Step 1: Convert the given vectors into x and y components.
For the vector 7 m/s [N30E]:
- The magnitude (length) of the vector is 7 m/s.
- The direction is represented as [N30E], which means a 30° angle counter-clockwise from the positive x-axis.
To find the x and y components, you can use basic trigonometry:
- The x component (horizontal) can be found using cosine: cos(30°) = adjacent/hypotenuse.
x component = 7 m/s * cos(30°).
- The y component (vertical) can be found using sine: sin(30°) = opposite/hypotenuse.
y component = 7 m/s * sin(30°).
For the vector 2 m/s [S17E]:
- The magnitude is 2 m/s.
- The direction is represented as [S17E], meaning a 17° angle clockwise from the positive x-axis.
To find the x and y components, you again use trigonometry:
- The x component is 2 m/s * cos(17°).
- The y component is 2 m/s * sin(17°).
Step 2: Add the x and y components separately.
Add the x components together: x_sum = x_1 + x_2.
Add the y components together: y_sum = y_1 + y_2.
Step 3: Combine the x_sum and y_sum to find the resultant vector.
To find the magnitude (length) of the resultant vector, use the Pythagorean theorem:
Resultant magnitude = √(x_sum² + y_sum²).
To find the direction of the resultant, use the inverse tangent (tan⁻¹) function:
Resultant direction = tan⁻¹(y_sum / x_sum). Be cautious of the quadrant in which the resultant lies.
By following these steps, you should be able to add the given vectors and find the resultant vector.