# Vector Problem ? Find windspeed

In summary, an airplane flying with a heading of N80°W and an airspeed of 680.0 km/h has a groundspeed of 650.0 km/h at a heading of N85°W. Using vector analysis, it can be determined that the windspeed is 65.3 km/h at a bearing of S19.82°E. An alternative method would be to draw a diagram and use basic geometry to solve the problem.
An airplane heads N80W with an airspeed of 680.0 km/h. Measurements made from the ground indicate that the plane’s groundspeed is 650.0 km/h at N85W. Find the windspeed and wind direction. [7 marks]

So a bearing of N80°W = 90+80 =170°
and a bearing of N85°W = 90+85 = 175°

Let a = vector indicating airplane's speed and direction.
Let w = vector indicating wind's speed and direction.
Let g = vector indicating plane's speed and direction from the ground

a + w = g
w = g - a
w = 650 (cos(175), sin(175)) - 680 (cos(170), sin(170))
w = (650cos(175)-680cos(170), 650sin(175)-680sin(170))
w = (22.142718, -61.429528)

|w| = √(22.142718^2 + 61.429528^2) = 65.298445

w = 65.298445 (cos(θ), sin(θ))

sin(θ) = -61.429528/65.298445 = -0.94075
cos(θ) = 22.142718/65.298445 = 0.3391
tan(θ) = -61.429528/22.142718

Since cos(θ) > 0 and sin(θ) < 0, then terminal point of θ is in quadrant IV
arctan has range between -90 and 90 (i.e. values in quadrant IV and quadrant I)
So we can just take arctan to find θ
(Otherwise, when θ is in QII or QIII, then we have to add 180 to arctan)

θ = arctan (-61.429528/22.142718) = -70.18

Changing this back into a bearing, we get S19.82°E

So windspeed = 65.3 km/h at S19.82°E

This this correct? Is there a simpler way to solve this problem.

An airplane heads N80W with an airspeed of 680.0 km/h. Measurements made from the ground indicate that the plane’s groundspeed is 650.0 km/h at N85W. Find the windspeed and wind direction. [7 marks]

So a bearing of N80°W = 90+80 =170°
and a bearing of N85°W = 90+85 = 175°

Let a = vector indicating airplane's speed and direction.
Let w = vector indicating wind's speed and direction.
Let g = vector indicating plane's speed and direction from the ground

a + w = g
w = g - a
w = 650 (cos(175), sin(175)) - 680 (cos(170), sin(170))
w = (650cos(175)-680cos(170), 650sin(175)-680sin(170))
w = (22.142718, -61.429528)

|w| = √(22.142718^2 + 61.429528^2) = 65.298445

w = 65.298445 (cos(θ), sin(θ))

sin(θ) = -61.429528/65.298445 = -0.94075
cos(θ) = 22.142718/65.298445 = 0.3391
tan(θ) = -61.429528/22.142718

Since cos(θ) > 0 and sin(θ) < 0, then terminal point of θ is in quadrant IV
arctan has range between -90 and 90 (i.e. values in quadrant IV and quadrant I)
So we can just take arctan to find θ
(Otherwise, when θ is in QII or QIII, then we have to add 180 to arctan)

θ = arctan (-61.429528/22.142718) = -70.18

Changing this back into a bearing, we get S19.82°E

So windspeed = 65.3 km/h at S19.82°E

This this correct? Is there a simpler way to solve this problem.

May I surgest you take a piece of paper and draw the situation like a simple geometric math problem! Then its often easier to solve.

## 1. What is a vector problem?

A vector problem involves finding the magnitude and direction of a vector, which is a quantity that has both size and direction. These problems are commonly encountered in physics and engineering.

## 2. How do you solve a vector problem?

To solve a vector problem, you need to break down the vector into its components, which are the parts of the vector that act in different directions. Then, you can use trigonometry or other mathematical methods to find the magnitude and direction of the vector.

## 3. What is the importance of finding windspeed in vector problems?

Windspeed is an important component in many vector problems, especially those involving motion or forces in the atmosphere. It helps determine the overall direction and strength of the vector in question.

## 4. What tools are used to find windspeed in vector problems?

In vector problems, windspeed can be found using tools such as a protractor, a compass, or a graphing calculator. These tools help measure angles and distances, which are necessary for solving vector problems.

## 5. How can I practice solving vector problems involving windspeed?

There are many resources available for practicing vector problems and specifically those involving windspeed. You can find worksheets, online tutorials, and practice problems in physics or engineering textbooks. Additionally, there are many interactive simulations and games available online that allow you to practice solving vector problems in a fun and engaging way.

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