An object has three component velocities: 5mi/hr toward the east, 12mi/hr toward the south, and 8mi/hr toward the northwest. Find the resultant velocity of the object.

For resultant velocity, I got 6mi/hr
I don't understand how to find the direction angle and direction (ex. N of E)
Thanks for your help!

Vr = 5 - 12i + 8[135o]

Vr = 5 - 12i + 8*Cos135 + i8*sin135
Vr = 5 - 5.66-12i + 5.66i=-0.66 - 6.34i
(Q3).

Tan Ar = Y/X = -6.34/-0.66 = 9.61083
Ar = 84o = Reference angle.
A = 84 + 180 = 264o CCW = 84o S. of W.

Vr = Y/sin A = -6.34/sin264 = 6.37 mi/h.
= Resultant velocity.

Note: Since the angle is in the 3rd quadrant, it is 84o S. of W. or 6o W. of
S.

Hallie is trying to win the grand prize on a game show. Should she try her luck by spinning a wheel with 6 equal sections labeled from 1 to 6 and hope she gets a 5, or should she roll two number cubes and hope she gets the same number on both cubes? Explain

Hallie is trying to win the grand prize on a game show. Should she try her luck by spinning a wheel with 6 equal sections labeled from 1 to 6 and hope she gets a 5, or should she roll two number cubes and hope she gets the same number on both cubes? Explain

Hallie is trying to win the grand prize on a game show. Should she try her luck by spinning a wheel with 6 equal sections labeled from 1 to 6 and hope she gets a 5, or should she roll two number cubes and hope she gets the same number on both cubes? Explain

To find the resultant velocity of the object, we need to determine both the magnitude (speed) and the direction of the resulting velocity vector.

Step 1: Break down the given velocities into their x and y components:

Velocity toward the east = 5 mi/hr, which can be represented as (5, 0) since there is no north or south component.
Velocity toward the south = 12 mi/hr, which can be represented as (0, -12) since there is no east or west component.
Velocity toward the northwest can be broken down into its x and y components using trigonometry. The northwest direction is a combination of the north and west directions. Let's call the x component v_x and the y component v_y. We can use the Pythagorean theorem and trigonometric ratios to find these components:
Magnitude of velocity toward northwest = √(8^2 + 8^2) = √(64 + 64) = √128 = 8√2 mi/hr

To find the x and y components, we can use the angle between the resultant velocity vector and the x-axis:
Angle = tan^(-1)(v_y / v_x) = tan^(-1)(-8 / -8) = tan^(-1)(1) = 45°

Using trigonometric ratios, we can find the x and y components of the velocity:
v_x = magnitude * cos(angle) = 8√2 * cos(45°) = 8√2 * (√2 / 2) = 8 mi/hr
v_y = magnitude * sin(angle) = 8√2 * sin(45°) = 8√2 * (√2 / 2) = 8 mi/hr

So, the velocity toward the northwest can be represented as (8, 8).

Step 2: Add the x and y components of all the velocities to find the resultant velocity:

Resultant velocity in the x-direction:
5 mi/hr (east) + 8 mi/hr (northwest) = 5 + 8 = 13 mi/hr (east)

Resultant velocity in the y-direction:
12 mi/hr (south) + 8 mi/hr (northwest) = -12 + 8 = -4 mi/hr (south)

Therefore, the resultant velocity of the object is (13, -4) mi/hr.

Step 3: Find the magnitude and direction of the resultant velocity:

Magnitude of the resultant velocity = √(13^2 + (-4)^2) = √(169 + 16) = √185 ≈ 13.6 mi/hr

To find the direction, we need the angle formed by the resultant velocity vector with the positive x-axis. We can again use trigonometry:

Angle = tan^(-1)(v_y / v_x) = tan^(-1)(-4 / 13) ≈ -17.7°

To express the direction in terms of "N of E" or "S of W", we can take the absolute value of the angle:

Absolute value of angle = |-17.7°| ≈ 17.7°

Since the angle is negative, the direction is to the south of the east. So, the direction of the resultant velocity is approximately S17.7°E.

Therefore, the resultant velocity of the object is approximately 13.6 mi/hr in the direction of S17.7°E.