Question
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Respond to the short answer questions below, showing all work.
1. A plane is traveling due west at 350 mph. A south wind is blowing at 30 mph. What is the resulting velocity of the plane?
2. Vectors u and v have an initial point of (-1,4).
a. Determine the component form of the vector v if its terminal point is (5, -9).
b. Determine the component form of the vector u if its terminal point is (7, -3).
c. Find the resultant of u and v and calculate its magnitude.
d. What is the direction angle of the resultant vector?
e. Calculate the dot product of u and v.
Respond to the short answer questions below, showing all work.
1. A plane is traveling due west at 350 mph. A south wind is blowing at 30 mph. What is the resulting velocity of the plane?
2. Vectors u and v have an initial point of (-1,4).
a. Determine the component form of the vector v if its terminal point is (5, -9).
b. Determine the component form of the vector u if its terminal point is (7, -3).
c. Find the resultant of u and v and calculate its magnitude.
d. What is the direction angle of the resultant vector?
e. Calculate the dot product of u and v.
Answers
Answered by
GPT 3.5
1. The resulting velocity of the plane can be found using vector addition. Let V be the velocity of the plane and W be the wind velocity. The resulting velocity R is given by R = V + W.
Let V = -350i, since the plane is traveling due west, and W = -30j, since the south wind is blowing in the negative y-direction.
R = V + W = -350i - 30j = -350i - 30j
Therefore, the resulting velocity of the plane is -350i - 30j mph.
2.
a. The component form of vector v can be found by calculating the differences of the x and y coordinates of the terminal and initial points.
v = (5-(-1), -9-4) = (6, -13)
b. The component form of vector u can be found in a similar manner.
u = (7-(-1), -3-4) = (8, -7)
c. To find the resultant vector, we add u and v component-wise.
u + v = (8+6, -7-13) = (14, -20)
The magnitude of the resultant vector R is given by |R| = sqrt(14^2 + (-20)^2) = sqrt(196 + 400) = sqrt(596) ≈ 24.43
d. The direction angle θ of the resultant vector is given by tan(θ) = (-20) / 14. Using trigonometry, we find θ ≈ -52.13 degrees.
e. The dot product of vectors u and v is given by u ∙ v = (8)(6) + (-7)(-13) = 48 + 91 = 139.
Let V = -350i, since the plane is traveling due west, and W = -30j, since the south wind is blowing in the negative y-direction.
R = V + W = -350i - 30j = -350i - 30j
Therefore, the resulting velocity of the plane is -350i - 30j mph.
2.
a. The component form of vector v can be found by calculating the differences of the x and y coordinates of the terminal and initial points.
v = (5-(-1), -9-4) = (6, -13)
b. The component form of vector u can be found in a similar manner.
u = (7-(-1), -3-4) = (8, -7)
c. To find the resultant vector, we add u and v component-wise.
u + v = (8+6, -7-13) = (14, -20)
The magnitude of the resultant vector R is given by |R| = sqrt(14^2 + (-20)^2) = sqrt(196 + 400) = sqrt(596) ≈ 24.43
d. The direction angle θ of the resultant vector is given by tan(θ) = (-20) / 14. Using trigonometry, we find θ ≈ -52.13 degrees.
e. The dot product of vectors u and v is given by u ∙ v = (8)(6) + (-7)(-13) = 48 + 91 = 139.