Vector a has a magnitude of 5 m and is directed east. vector b has magnitude of 4m and is directed 35 degrees west of north.
A)What is the magnitude and direction of vector a + vector b
b)What is the magnitude and direction of vector b- vector a?
I should have gotten
a)4.2m at 40 degrees east of north
b)8m at 24 degrees north of west
I am confused by this problem
Change vector b to N,E coordinates.
I get B= -2.3E + 3.3N
Now add them for part a.
A+ B= 5E -2.3E + 3.3N and that converts to the answer given.
For part b, subtract them.
B-A= -2.3E + 3.3N - 5E = -7E + 3.3N and that converts to the answer given.
To find the magnitude and direction of vector a + vector b, we need to add the respective components of the vectors and then calculate the resultant magnitude and direction.
Let's first convert vector b to east (E) and north (N) coordinates to simplify the addition:
Vector b has a magnitude of 4 m and is directed 35 degrees west of north. To convert this to E and N coordinates, we need to find the horizontal and vertical components:
Horizontal component of b = b * cos(35°) = 4 * cos(35°) = 4 * 0.819 = 3.276 ≈ 3.3
Vertical component of b = b * sin(35°) = 4 * sin(35°) = 4 * 0.574 = 2.296 ≈ 2.3
So, vector b in E, N coordinates is:
Vector b = 3.3E + 2.3N
Now, let's add vector a and vector b:
Vector a has a magnitude of 5 m and is directed east.
To add vector a and vector b, we add their corresponding components:
Horizontal component of a + b = 5 + 3.3 = 8.3
Vertical component of a + b = 0 + 2.3 = 2.3
So, the resultant vector a + b in E, N coordinates is:
Vector a + b = 8.3E + 2.3N
Now, let's find the magnitude and direction of the resultant vector a + b:
Magnitude of a + b = √(8.3^2 + 2.3^2) = √(68.89 + 5.29) ≈ √74.18 ≈ 8.6 m (rounded to one decimal place)
Direction of a + b = tan^(-1)(2.3/8.3) ≈ 16.5° (measured from the positive E-axis)
Therefore, the magnitude and direction of vector a + vector b are approximately 8.6 m at 16.5° east of north.
For part b, we need to find the magnitude and direction of vector b - vector a.
Since vector b is in E, N coordinates and vector a is directed east (E) with no vertical component, we can directly subtract the respective components:
Horizontal component of b - a = 3.3 - 5 = -1.7
Vertical component of b - a = 2.3 - 0 = 2.3
So, the resultant vector b - a in E, N coordinates is:
Vector b - a = -1.7E + 2.3N
Now, let's find the magnitude and direction of the resultant vector b - a:
Magnitude of b - a = √((-1.7)^2 + 2.3^2) = √(2.89 + 5.29) ≈ √8.18 ≈ 2.9 m (rounded to one decimal place)
Direction of b - a = tan^(-1)(2.3/-1.7) ≈ -51.3° (measured from the positive E-axis)
Therefore, the magnitude and direction of vector b - vector a are approximately 2.9 m at 51.3° south of west.
To solve the problem, you are given the magnitude and direction of vectors a and b. We can break down the steps to find the magnitude and direction of vector a + vector b and vector b - vector a.
a) To find the magnitude and direction of vector a + vector b, you first need to convert vector b to its North and East (N,E) coordinates. Vector b is given as having a magnitude of 4m and is directed 35 degrees west of north.
To convert this to N,E coordinates, we can use trigonometry. Since vector b is directed 35 degrees west of north, this means it is 35 degrees west of the positive y-axis.
To find the North component of vector b, we can use the sine function: North Component = 4m * sin(35°), which is approximately 2.314m.
To find the East component of vector b, we can use the cosine function: East Component = 4m * cos(35°), which is approximately 3.276m.
Therefore, vector b can be written as B = -3.276E + 2.314N (note the negative sign for the East component since it is directed west).
Now, to find the vector a + vector b, you can simply add the corresponding components together.
Adding the East components: 5m + (-3.276m) = 1.724m.
Adding the North components: 0m + 2.314m = 2.314m.
So, vector a + vector b has a magnitude of √((1.724m)^2 + (2.314m)^2) ≈ 2.983m.
To find the direction of the vector, we can use inverse tangent: angle = atan(2.314m / 1.724m) ≈ 51.1°.
Therefore, the magnitude and direction of vector a + vector b is approximately 2.983m at 51.1 degrees east of north.
b) To find the magnitude and direction of vector b - vector a, we can use the same process but subtract the corresponding components.
The East component of vector b - vector a would be: -3.276m - 5m = -8.276m.
The North component of vector b - vector a would be: 2.314m - 0m = 2.314m.
So, vector b - vector a has a magnitude of √((-8.276m)^2 + (2.314m)^2) ≈ 8.494m.
To find the direction of the vector, we can use inverse tangent: angle = atan(2.314m / -8.276m) ≈ -14.94°.
Note that we have a negative angle because the vector is pointing west of north. To convert it to a positive angle counterclockwise from the positive x-axis, we can add 360° to the negative angle: 360° - 14.94° ≈ 345.06°.
Therefore, the magnitude and direction of vector b - vector a is approximately 8.494m at 345.06 degrees north of west.
I hope this explanation helps clarify the problem for you! Let me know if you have any further questions.