Vector a has a magnitude of 5 m and is directed esast. Vector b has a magnitude of 4 m and is directed 35 degrees west of north. What are the (a) magnitude and (b) the direction of vector a + vector b?
What are the (c)magnitude and (e)direction of vector b - vecotr a?
(f) Draw a vector diagram for each combination.
My teacher ran out of time trying to explain this problem with us. And we are having a test on this kind of problem too. I was wondering how my teacher got these answers. All of the ways I have tried getting these answers I keep on getting the wrong numbers. Please help me.
a)4.2m
b)40 degrees east of north
c)8 m
d) 24 degrees north of west
e) How would I draw this?
Vector A is 5E
Vector B is 4 at 35W of N.
Change vector B to E,N
Vector B= 4N cos(360-35) + 4Esin(360-35)
This angle change is a simple way to keep it straight (cos from N clockwise is N component, Sin from N clockwise is E component).
Then convert Vector B
Vector B= 3.28N -2.29E
So, Vector A + B= 5E + 3.28N -2.29E
=2.71E + 3.28N
Magnitude = sqrt (2.71^2 + 3.28^2)
angle from N is arctan 2.17/3.28
I am not getting exactly the teachers answers, check my work, I or she could have made an error.
Let's go step by step to find the correct answers:
Given information:
Vector A: magnitude = 5 m, direction = east (E)
Vector B: magnitude = 4 m, direction = 35 degrees west of north
To find the magnitude and direction of A + B:
To add vectors, we first need to resolve them into their x and y components.
Vector A: 5 m, east (E) direction
- x-component: 5 cos(0 degrees) = 5
- y-component: 5 sin(0 degrees) = 0
Vector B: 4 m, 35 degrees west of north
- To convert the direction to an east (E) and north (N) coordinate system, we subtract 35 degrees from 90 degrees (north).
- The new direction is 55 degrees west of east.
- x-component: 4 cos(55 degrees) ≈ 2.417
- y-component: 4 sin(55 degrees) ≈ 3.144
Now, we can add the x and y components of A and B separately:
x-component(A + B) = 5 + 2.417 ≈ 7.417
y-component(A + B) = 0 + 3.144 ≈ 3.144
To find the magnitude of A + B:
Magnitude(A + B) = sqrt((7.417)^2 + (3.144)^2) ≈ sqrt(54.89 + 9.87) ≈ sqrt(64.76) ≈ 8.06 m (rounded to two decimal places)
To find the direction of A + B:
The angle from the north can be calculated as:
Angle(A + B) = arctan(3.144/7.417) ≈ 23.87 degrees
Since the vector is directed east (E) of north, we need to subtract this angle from 90 degrees:
Direction(A + B) = 90 - 23.87 ≈ 66.13 degrees (rounded to two decimal places)
Hence, the answers are:
(a) The magnitude of A + B is approximately 8.06 m.
(b) The direction of A + B is approximately 66.13 degrees east of north.
Now, let's move on to finding the magnitude and direction of B - A:
To subtract vectors, we again resolve them into their x and y components.
Vector A: 5 m, east (E) direction
- x-component: 5 cos(0 degrees) = 5
- y-component: 5 sin(0 degrees) = 0
Vector B: 4 m, 35 degrees west of north
- x-component: 4 cos(35 degrees) ≈ 3.276
- y-component: 4 sin(35 degrees) ≈ 2.286
Now, we can subtract the x and y components of A and B separately:
x-component(B - A) = 3.276 - 5 ≈ -1.724
y-component(B - A) = 2.286 - 0 ≈ 2.286
To find the magnitude of B - A:
Magnitude(B - A) = sqrt((-1.724)^2 + (2.286)^2) ≈ sqrt(2.976 + 5.239) ≈ sqrt(8.215) ≈ 2.87 m (rounded to two decimal places)
To find the direction of B - A:
The angle from the north can be calculated as:
Angle(B - A) = arctan(2.286/-1.724) ≈ -52.64 degrees (rounded to two decimal places)
Since this angle is negative, we need to add 180 degrees to get the direction in the compass system:
Direction(B - A) = -52.64 + 180 ≈ 127.36 degrees
Hence, the answers are:
(c) The magnitude of B - A is approximately 2.87 m.
(d) The direction of B - A is approximately 127.36 degrees (rounded to two decimal places).
For drawing the vector diagrams:
To draw vector A, just draw an arrow of length 5 units (representing 5 m) in the east (right) direction.
To draw vector B, start from the same point as the start of vector A and draw an arrow of length 4 units at an angle of 35 degrees west of north.
For vector A + B, start from the tail of vector A and draw an arrow of length 8.06 units in the direction of 66.13 degrees east of north.
For vector B - A, start from the tail of vector A and draw an arrow of length 2.87 units in the direction of 127.36 degrees.
I hope this explanation helps you understand how to solve this problem and obtain the correct answers. Good luck with your test!