three vectos are shown in fig. 3-32. their magnitudes are given in arbitrary units. Determine the sum of the three vectors. Give the resultant in terms of (a) components, (b) magnitude and angle with the x axis.
(3-32 --the figure has three rays all starting at the origin. Ray A = 44.0 and is 28degress from the x axis in quodrent 1. Ray B=26.5 and is 56degress from the x axis in quodrent 2. Ray C=31.0 and is on the y axis between quodrent 3 and 4.)
break up all three vectors into x,y components. then add the x's, and the y's.
That is the resultant vector.
a)4
b)4
C)3
A0
b)
c0
To determine the sum of the three vectors, you need to find the components of each vector and add them together.
(a) Finding the components:
To find the components of each vector, you can use the magnitudes and the angles given. The components of a vector can be calculated using the following formulas:
- x-component = magnitude * cos(angle)
- y-component = magnitude * sin(angle)
Let's calculate the components for each vector:
Vector A:
Magnitude = 44.0
Angle with the x-axis = 28 degrees
x-component of A = 44.0 * cos(28°)
y-component of A = 44.0 * sin(28°)
Vector B:
Magnitude = 26.5
Angle with the x-axis = 56 degrees
x-component of B = 26.5 * cos(56°)
y-component of B = 26.5 * sin(56°)
Vector C:
Magnitude = 31.0
Angle with the x-axis = 90 degrees (as it lies on the y-axis)
x-component of C = 0 (since it lies on the y-axis)
y-component of C = 31.0
(b) Calculating the sum of components:
To find the sum of the components, simply add the x-components and y-components separately:
x-component of the resultant vector = Sum of x-components of A, B, and C
y-component of the resultant vector = Sum of y-components of A, B, and C
(c) Finding the magnitude and angle of the resultant vector:
To find the magnitude of the resultant vector, you can use the Pythagorean theorem:
magnitude of the resultant vector = sqrt((x-component of the resultant vector)^2 + (y-component of the resultant vector)^2)
To find the angle of the resultant vector with the x-axis, you can use the inverse tangent function:
angle with x-axis = atan2(y-component of the resultant vector, x-component of the resultant vector)
Now, you can follow these steps to calculate the sum in terms of components, magnitude, and angle with respect to the x-axis.