Final answer:
To find the trigonometric form of vector \(\overline{PQ}\) between points P(-5,11) and Q(-16,4), one must calculate the vector's components, magnitude, and direction angle. The direction angle must account for the vector being in the third quadrant. The correct trigonometric form is 13.038cos(212.471°)i+13.038sin(212.471°)j.
Step-by-step explanation:
To determine which option represents the vector t=\(\overline{PQ}\) in trigonometric form, where P(-5,11) and Q(-16,4), we first calculate the components of the vector t by subtracting the coordinates of P from Q:
- tx = Qx - Px = -16 - (-5) = -11
- ty = Qy - Py = 4 - 11 = -7
The magnitude of vector t can be found using the Pythagorean theorem:
|t| = \(\sqrt{t_{x}^{2} + t_{y}^{2}}\) = \(\sqrt{(-11)^{2} + (-7)^{2}}\) = \(\sqrt{121 + 49}\) = \(\sqrt{170}\) = 13.038
Next, we need to find the direction angle of vector t. The angle θ formed with the positive x-axis can be found using the arctangent function:
θ = tan-1\(\frac{t_{y}}{t_{x}}\) = tan-1\(\frac{-7}{-11}\) = tan-1\(\frac{7}{11}\). After calculation, θ is approximately 32.471°.
Since vector t is in the third quadrant (both x and y components are negative), we must add 180° to θ to obtain the correct direction angle, which is 212.471°.
Now that we have the magnitude and the direction, we can express vector t in trigonometric form as:
t = |t| cos(θ) i + |t| sin(θ) j
In this case, this translates to:
t = 13.038 cos(212.471°) i + 13.038 sin(212.471°) j
The correct option that represents vector t=\(\overline{PQ}\) in trigonometric form is therefore option (b).