vector a is 8 ft., vector b is 2 ft, and theta is at 41 degrees. find the resultant and the direction.
I drew a diagram that shows a connecting with b (head to tail), and they make the 41 degrees. I use law of cosines and got about 6ft, which doesn't sound right to me.
my work:
r^2 = 8^2 + 2^2 - 2(8)(2)cos41
r^2 = 43.849
r = 6.621 ?
I was thinking the resultant would be larger than vector a. Also, how do you find the direction (angle) of the resultant?
Thanks!
Usually the given angle is the angle between the two vectors, so after you place them end to end, I saw a triangle with sides 8 and 2 and a contained angle of 139º ( I placed the vector a horizontally)
Now by the cosine law,
R^2 = 64 + 4 - 2(8)(2)cos139
R = 9.6 (9.5995)
Let ß be the angle between the resultant R and the vector of length 8
sinß/2 = sin139/9.5995
ß = 7.856º degrees up from the horizontal.
To find the resultant magnitude and direction, you correctly used the Law of Cosines. However, there seems to be a mistake in your calculation.
To calculate the magnitude of the resultant vector, you can use the following equation:
r^2 = a^2 + b^2 - 2ab * cos(theta)
Substituting the given values:
r^2 = 8^2 + 2^2 - 2 * 8 * 2 * cos(41)
r^2 = 64 + 4 - 32 * cos(41)
r^2 = 68 - 32 * 0.7539
r^2 = 68 - 24.1248
r^2 = 43.8752
To calculate the resultant magnitude, take the square root of both sides:
r = sqrt(43.8752)
r ≈ 6.62 ft (rounded to two decimal places)
So, your calculated value of 6.621 ft is indeed correct, but there was a slight rounding error.
Now, to determine the direction (angle) of the resultant vector, you can use the Law of Sines or the Law of Tangents. Both methods will yield the same angle, but let's use the Law of Sines.
sin(theta) / b = sin(theta + angle) / r
Substituting the given values:
sin(41) / 2 = sin(theta + angle) / 6.62
Cross-multiplying:
sin(theta + angle) = 6.62 * sin(41) / 2
To find the angle, you can use inverse sine:
theta + angle = arcsin(6.62 * sin(41) / 2)
theta + angle ≈ arcsin(6.62 * 0.6561)
theta + angle ≈ arcsin(4.34)
theta + angle ≈ 77.26 degrees
To find the angle (direction) of the resultant vector, subtract theta from the obtained angle:
angle ≈ 77.26 - 41
angle ≈ 36.26 degrees
Therefore, the magnitude of the resultant vector is approximately 6.62 ft, and the direction is approximately 36.26 degrees from the original vector a.