dissplacement vector A is 4.5 km North, vector B is 3.0 km 50 degrees N of W. Find their resultant

Hey umm, isnt h=7.06 instead of 7.6(?)

vector A is 4.5 Km N, this means that for this vector we are not going to have an x- coordinate, we are only going to have a y- coordinate , since this vector is going N then the y- cordinate is postive number, and the y-coordinate for this vector will be + 4.5, while the x-vector is going to be 0

vector B is 3 Km 50 degrees N of W, this simply means 3 Km W50N , for vector B, since the object is moving both in the x- direction and the y- direction then we are going to have an x- cordinate and a y- coordinate for this vector, also since the directions that this vector has are N and W , threrefore this means that we are going to have a negative x- coordinate ( because the object is going to the west in terms of the horizontal direction) and a postive y- coordinate( since the object is going north in terms of the vertical direction)
in order to find coordinates for the second vector in terms of x and y: (- 3x cos50 , +3 sin50)
= (-1.928, 2.298)
then we are going to add both vectore together
( 0,4.5) +
(-1.928, 2.298)
=( -1.928, 6.798)
we are going to use the pythageorm theorm to find the resultant displacment, we are going to call it h
h^2=x^2+y^2
h^2= (-1.928)^2 + (6.798)^2
h^2= 49.931
h= 7.06 Km
then we know that the oject travelled a resultant displacment 7.06 Km, but we need to know in which direction did the resultant vector traveled.
from the sum of the 2 vectors: (-1.928,6.798) we know that we have a negative x- cordinate which means we are going west horizontaly , for the y-coordinate we have a postive value which means we are going north verticaly) we are going to use H to find the angle in which the resultant vector is sin^-1( 6.728/7.60)= 62.72
therfore the final resultant is 7.60 Km w62.2 N

Well, if A is going North and B is going 50 degrees N of W, then I have to say they really have their own goals in mind! It's like they're not even coordinating their efforts. Classic case of miscommunication.

But here's the deal: We can't directly add vectors that are not going in the same direction. So, what we have to do is break vector B into horizontal and vertical components.

First, let's find the horizontal component of vector B. Since it's pointing 50 degrees N of W, it's also pointing 40 degrees W of N (because angles on a straight line always add up to 180 degrees).

Next, we can find the horizontal and vertical components using trigonometry. The horizontal component of B would be 3.0 km multiplied by the cosine of 40 degrees (since cosine gives the adjacent side over the hypotenuse).

So, the horizontal component of B is 3.0 km times cos(40 degrees), which is approximately 2.31 km.

The vertical component of B would be 3.0 km multiplied by the sine of 40 degrees (since sine gives the opposite side over the hypotenuse).

So, the vertical component of B is 3.0 km times sin(40 degrees), which is approximately 1.92 km.

Now that we have broken down vector B into its horizontal and vertical components, we can add them to vector A.

The horizontal component of the resultant vector would be 4.5 km plus 2.31 km (which is the horizontal component of B), giving us approximately 6.81 km.

The vertical component of the resultant vector would be 1.92 km plus 0 km (since vector A has no vertical component), giving us approximately 1.92 km.

Now we have the horizontal and vertical components of the resultant vector, we can use the Pythagorean theorem to find the magnitude of the resultant.

The magnitude of the resultant vector is the square root of the sum of the squares of the horizontal and vertical components.

So, the magnitude of the resultant vector is approximately sqrt((6.81 km)^2 + (1.92 km)^2) = sqrt(51.37 km^2 + 3.69 km^2) = sqrt(55.06 km^2) = 7.42 km.

The direction of the resultant vector can be found using trigonometry as well. The angle the resultant vector makes with the horizontal axis can be found as the inverse tangent of the vertical component over the horizontal component.

So, the angle of the resultant vector is approximately atan(1.92 km / 6.81 km) = atan(0.282) = 15.84 degrees.

Therefore, the resultant vector has a magnitude of approximately 7.42 km and makes an angle of approximately 15.84 degrees with the horizontal axis.

To find the resultant of displacement vectors A and B, we can use vector addition. Here's how you can do it step by step:

Step 1: Draw a diagram
Start by drawing a coordinate system with a North-South (vertical) and East-West (horizontal) reference. Draw vector A as a straight arrow pointing 4.5 km upwards (North) from the origin. Then, draw vector B as a straight arrow pointing 3.0 km in the direction that is 50 degrees North of West (counterclockwise from the West direction).

Step 2: Resolve vector B
Resolve vector B into its North-South (y-component) and East-West (x-component) parts. To do this, use trigonometry. Since the angle is measured counterclockwise from the West direction, subtract 50 degrees from 180 degrees to get 130 degrees, which gives us the reference angle. Then, use the sine and cosine functions to find the y-component (North) and x-component (West).

y-component (North): B * sin(130)
x-component (West): B * cos(130)

Step 3: Add the components
Combine the North components (y-components) of A and B and the West components (x-components) of A and B.

North component: A + B * sin(130)
West component: -B * cos(130)

Step 4: Find the magnitude and direction of the resultant
Use the Pythagorean theorem to find the magnitude of the resultant vector.

Magnitude: √(North component^2 + West component^2)

To find the direction of the resultant vector, use inverse trigonometric functions. The direction will be the angle measured counterclockwise from the West direction.

Direction: arctan(West component / North component)

Step 5: Calculate the values
Substitute the values of A (4.5 km), B (3.0 km), and the components calculated in Step 2 and Step 3 into the equations.

North component: 4.5 km + 3.0 km * sin(130)
West component: -3.0 km * cos(130)

Step 6: Calculate the magnitude and direction
Plug the values of the North and West components calculated in Step 5 into the equations to find the magnitude and direction of the resultant.

Magnitude: √((4.5 km + 3.0 km * sin(130))^2 + (-3.0 km * cos(130))^2)

Direction: arctan((-3.0 km * cos(130)) / (4.5 km + 3.0 km * sin(130)))

By following these steps, you can find the resultant of displacement vectors A and B.

7.07