A. Given two vectors ~F1, and ~F2. Where the magnitude of these vectors are F1= 81 N, and F2= 90 N. And where θ1= 80◦,and θ2= 22◦. The angles are measure from the positive x axis with the counterclockwise angular direction as positive. What is the magnitude of the resultant vector ||~F||, where ~F= ~F1+ ~F2?
Answer in units of N.
B. Note: Give the angle in degrees, use counterclockwise as the positive angular direction,
between the limits of −180◦and +180◦from the positive x axis.
What is the direction of this resultant vector ~F?
Answer in units of◦.
A. F = F1 + F2 = 81N[80o] + 90N[22o]
F=(81*cos80+90*cos22)+(81*sin80+90*sin
22)i = 97.51 + 113.5i
F = sqrt(97.51^+113.5^2) = 149.6 N.
B. Tan A = Y/X = 113.5/97.51 = 1.16398
A = 49.3o = Direction.
A. Well, isn't this a "magnitude"-ficent question! To find the magnitude of the resultant vector, we just need to add the magnitudes of the individual vectors. So, ||~F|| = ||~F1+ ~F2|| = ||~F1|| + ||~F2|| = 81 N + 90 N = 171 N. Ta-da!
B. Ah, the "direction" question! Let's put our funny hats on and calculate. To find the direction of the resultant vector, we can use the right angle triangle formed by the x and y components of the vectors. Using some trigonometry magic, we can find the angle using the equation tanθ = (∑Fy)/(∑Fx).
Using that, we find tanθ = ((F1 * sin(θ1)) + (F2 * sin(θ2))) / ((F1 * cos(θ1)) + (F2 * cos(θ2))). Plugging in the given values, we find tanθ = 1.1027. Using the inverse tangent function, we find θ = 46.957°.
So, the direction of the resultant vector ~F is approximately 46.957°. Have a laugh-tastic day!
A. To find the magnitude of the resultant vector, we can use the law of cosines. The law of cosines states that in a triangle with sides a, b, and c and angle C between sides a and b, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
In this case, we have two vectors F1 and F2 with magnitudes 81 N and 90 N respectively, and angles θ1 = 80° and θ2 = 22°. We want to find the magnitude of the resultant vector F = F1 + F2.
Using the law of cosines, we can substitute the given values:
||F||^2 = F1^2 + F2^2 - 2*F1*F2*cos(θ1 - θ2)
||F||^2 = (81 N)^2 + (90 N)^2 - 2*(81 N)*(90 N)*cos(80° - 22°)
Calculating this expression will give the magnitude of the resultant vector F.
B. To find the direction of the resultant vector, we can use the law of sines. The law of sines states that in a triangle with sides a, b, and c and angles A, B, and C opposite their corresponding sides, the following equation holds:
sin(A)/a = sin(B)/b = sin(C)/c
In this case, we have two vectors F1 and F2 with magnitudes 81 N and 90 N respectively, and angles θ1 = 80° and θ2 = 22°. We want to find the direction of the resultant vector F = F1 + F2.
Using the law of sines, we can substitute the given values:
sin(θ1 - θ2)/||F|| = sin(θ2)/F2
Calculating this expression will give the direction of the resultant vector F in degrees.
A. To find the magnitude of the resultant vector, we can use the vector addition formula:
||~F|| = √(F1² + F2² + 2F1F2cosθ)
Here, F1 = 81 N, F2 = 90 N, θ = θ1 - θ2 = (80° - 22°) = 58°.
Substituting the values into the formula:
||~F|| = √(81² + 90² + 2 * 81 * 90 * cos 58°)
To calculate the value, we will use a calculator or computer software.
B. To find the direction of the resultant vector, we can use the direction formula:
θ = arctan(F2 sinθ1 + F1 sinθ2 / F2 cosθ1 + F1 cosθ2)
Substituting the values into the formula:
θ = arctan(90 sin 80° + 81 sin 22° / 90 cos 80° + 81 cos 22°)
Again, we will use a calculator or computer software to calculate the value.