Given that P = (-4, -5) and Q = (2,3), express →PQ in the form (k,θ). where k is the magnitude and θ the bearing.
Answer Details
To find →PQ, we need to subtract the coordinates of point P from those of point Q, giving us:
→PQ = Q - P = (2-(-4), 3-(-5)) = (6, 8)
The magnitude, or length, of →PQ is found using the distance formula:
k = √(6² + 8²) = √100 = 10
To find the bearing θ, we use trigonometry. The tangent of θ is the ratio of the opposite side (the change in y-coordinates) to the adjacent side (the change in x-coordinates):
tan θ = 8/6 = 4/3
Using a calculator, we can find that θ is approximately 53.13º. However, we need to adjust this value depending on which quadrant →PQ lies in. Since both x and y are positive, →PQ lies in the first quadrant, so we don't need to make any adjustments. Therefore:
θ = 53.13º
So the vector →PQ can be expressed in the form (k,θ) as:
(10 units, 53.13º)
Therefore, the correct answer is (a) (10 units, 053º).