How can one find the magnitude and the directions of a resultant if Vector A is 35N @ 60' NE, and Vector B is 50N @ 20' NW?
A = 35N @ 60deg,NE = 35N. @ 60deg,CCW.
B = 50N. @ 20deg,NW = 50N @ 160deg,CCW.
X=hor. = 35cos60 + 50cos160 = -29.48N.
Y = ver. = 35sin60 + 50sin160 = 47.4N.
R^2 = X^2 + Y^2,
R^2 = (-29.48)^2 + (47.4)^2 = 3117,
R = 55.8N.
tanA = Y/X = 47.4 / -29.48 = -1.60787,
A = -58.1 Deg.,CW.
A = -58.1 + 180 = 121.9 Deg,CCW. =
Direction.
R = 55.8N. @ 121.9 Deg.
To find the magnitude and direction of the resultant vector, you need to add Vector A and Vector B together using vector addition. The steps to find the resultant are as follows:
Step 1: Convert the given vectors into their horizontal (X) and vertical (Y) components:
Vector A:
Magnitude = 35 N
Direction = 60 degrees NE
To convert Vector A into its X and Y components:
X-component = Magnitude * cos(Direction)
Y-component = Magnitude * sin(Direction)
X-component of Vector A:
= 35 N * cos(60 degrees)
= 35 N * 0.5
= 17.5 N
Y-component of Vector A:
= 35 N * sin(60 degrees)
= 35 N * √(3)/2
≈ 30.43 N
Vector B:
Magnitude = 50 N
Direction = 20 degrees NW
To convert Vector B into its X and Y components:
X-component = Magnitude * cos(Direction)
Y-component = Magnitude * sin(Direction)
X-component of Vector B:
= 50 N * cos(20 degrees)
≈ 46.02 N
Y-component of Vector B:
= 50 N * sin(20 degrees)
≈ 17.09 N
Step 2: Add the X-components and Y-components together to find the resultant X and Y components:
X-component of the resultant:
= X-component of Vector A + X-component of Vector B
= 17.5 N + 46.02 N
≈ 63.52 N
Y-component of the resultant:
= Y-component of Vector A + Y-component of Vector B
= 30.43 N + 17.09 N
≈ 47.52 N
Step 3: Use the Pythagorean theorem to find the magnitude of the resultant vector:
Magnitude of the resultant:
= √((X-component of the resultant)^2 + (Y-component of the resultant)^2)
= √((63.52 N)^2 + (47.52 N)^2)
≈ 79.91 N
Step 4: Use trigonometry to find the direction (angle) of the resultant vector:
Direction of the resultant:
= arctan((Y-component of the resultant) / (X-component of the resultant))
= arctan(47.52 N / 63.52 N)
≈ 36.87 degrees
The magnitude of the resultant vector is approximately 79.91 N, and the direction is approximately 36.87 degrees.
To find the magnitude and direction of the resultant of two vectors, Vector A and Vector B, you can use vector addition. Here are the steps to follow:
Step 1: Convert the given vectors into their horizontal and vertical components.
Vector A:
Magnitude = 35N
Direction = 60° NE
Horizontal Component = Magnitude * cos(Direction)
Vertical Component = Magnitude * sin(Direction)
Vector B:
Magnitude = 50N
Direction = 20° NW
Horizontal Component = Magnitude * cos(Direction)
Vertical Component = Magnitude * sin(Direction)
Step 2: Add the horizontal and vertical components of Vector A and Vector B separately.
Horizontal Component of Resultant = Sum of Horizontal Components of Vector A and Vector B
Vertical Component of Resultant = Sum of Vertical Components of Vector A and Vector B
Step 3: Determine the magnitude of the resultant vector using the Pythagorean theorem.
Magnitude of Resultant = sqrt((Horizontal Component of Resultant)^2 + (Vertical Component of Resultant)^2)
Step 4: Calculate the direction of the resultant vector using trigonometry.
Direction (angle) of Resultant = atan(Vertical Component of Resultant / Horizontal Component of Resultant)
So, to find the magnitude and direction of the resultant vector, you need to evaluate the values obtained from these calculations.