Examples, solutions, videos, worksheets, games and activities to help PreCalculus students learn to solve navigation problems using vectors.

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**Navigation Problems**

The word problems encountered most often with vectors are navigation problems. These navigation problems use variables like speed and direction to form vectors for computation. Some navigation problems ask us to find the groundspeed of an aircraft using the combined forces of the wind and the aircraft. For these problems it is important to understand the resultant of two forces and the components of force.

**How we measure airspeed, wind, groundspeed, and other variables in navigation?**
**How to solve application problems using vectors?**

This video explains 2 basic application problems involving vectors.

Example 1:

A ship leaves port on a bearing of 28° and travels 7.5 miles. The ship then turns due east and travels 4.1 miles. How far is the ship from the port and what is its bearing?

Example 2:

Two tow trucks are pulling on a truck stuck in the mud. Tow truck #1 is pulling with a force of 635 lbs at 51° from the horizontal while tow truck #2 is pulling with a force of 592 lbs at 39° from the horizontal. What is the magnitude and direction of the resultant force?

**Word Problems Involving Velocity or Other Forces (Vectors)**

In this problem we do a word problem involving the bearing (direction) of a boat.

Example:

A boat is traveling at a speed of 30 mph. The vector that represents the velocity is 15<√2, -√2>. What is the bearing of the boat?

**Word Problems Involving Velocity or Other Forces (Vectors)**

In this problem we are given the bearing and velocity of a plane and the bearing and velocity of the wind; we want to find out the actual velocity of the plane after taking the wind into consideration

Example:

A plane leaves the airport on a bearing 45° traveling a 400 mph. The wind is blowing at a bearing of 135° at a speed of 40 mph. What is the actual velocity of the plane?

** Word Problems Involving Velocity or Other Forces (Vectors)**

In this problem we are given the force required to keep a box from sliding down a ramp. We want to know the level of inclination of the ramp.

Example:

A 700 lb force just keeps a 4000 lb box from sliding down a ramp. What is the angle of inclination of the ramp?

**Airplane and Wind Vector Word Problems**

Example:

An airplane is flying in the direction 15° North of East at 550 mph. A wind is blowing in the direction 15° South of East at 45 mph.

a) Find the component form of the velocity of the airplane and the wind.

b) Find the actual speed ("ground speed") and direction of the airplane.

Related topics:

More PreCalculus Lessons

More Lessons on Vectors

The word problems encountered most often with vectors are navigation problems. These navigation problems use variables like speed and direction to form vectors for computation. Some navigation problems ask us to find the groundspeed of an aircraft using the combined forces of the wind and the aircraft. For these problems it is important to understand the resultant of two forces and the components of force.

This video explains 2 basic application problems involving vectors.

Example 1:

A ship leaves port on a bearing of 28° and travels 7.5 miles. The ship then turns due east and travels 4.1 miles. How far is the ship from the port and what is its bearing?

Example 2:

Two tow trucks are pulling on a truck stuck in the mud. Tow truck #1 is pulling with a force of 635 lbs at 51° from the horizontal while tow truck #2 is pulling with a force of 592 lbs at 39° from the horizontal. What is the magnitude and direction of the resultant force?

In this problem we do a word problem involving the bearing (direction) of a boat.

Example:

A boat is traveling at a speed of 30 mph. The vector that represents the velocity is 15<√2, -√2>. What is the bearing of the boat?

In this problem we are given the bearing and velocity of a plane and the bearing and velocity of the wind; we want to find out the actual velocity of the plane after taking the wind into consideration

Example:

A plane leaves the airport on a bearing 45° traveling a 400 mph. The wind is blowing at a bearing of 135° at a speed of 40 mph. What is the actual velocity of the plane?

In this problem we are given the force required to keep a box from sliding down a ramp. We want to know the level of inclination of the ramp.

Example:

A 700 lb force just keeps a 4000 lb box from sliding down a ramp. What is the angle of inclination of the ramp?

Example:

An airplane is flying in the direction 15° North of East at 550 mph. A wind is blowing in the direction 15° South of East at 45 mph.

a) Find the component form of the velocity of the airplane and the wind.

b) Find the actual speed ("ground speed") and direction of the airplane.

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