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## Precalculus

### Course: Precalculus>Unit 6

Lesson 9: Vectors word problems

# Vectors: FAQ

## What is a vector?

A vector is a mathematical object that has both a magnitude (length) and a direction. Vectors can be used to represent physical quantities such as velocity, force, and acceleration.

## What are vector components?

Vector components are the horizontal and vertical parts of a vector. They can be found using trigonometric functions and the magnitude and direction of the vector.

## How do you add and subtract vectors?

There are a few different ways to add and subtract vectors. One way is to use vector components: you add or subtract the corresponding horizontal and vertical components to find the components of the resulting vector. Another way is to use vector magnitude and direction: you convert both vectors into this form, add or subtract the magnitudes, and use trigonometry to find the direction of the resulting vector.

## How do you multiply a vector by a scalar?

Multiplying a vector by a scalar (a number) changes its magnitude but not its direction. To multiply a vector by a scalar, you multiply each of its components by the scalar. For example, if v, with, vector, on top, equals, left parenthesis, 3, comma, 4, right parenthesis and k, equals, 2, then k, v, with, vector, on top, equals, left parenthesis, 6, comma, 8, right parenthesis.

## Where are vectors used in the real world?

Vectors are used in a wide variety of fields. In physics, vectors are used to describe motion (velocity, acceleration, force), while in engineering they are used to understand and design structures. Vectors are also used in computer graphics, where they are used to create and transform shapes.

## Want to join the conversation?

• What is an example of vectors being used in the real world?
(1 vote)
• Vectors have numerous applications in the real world, including:

Navigation: Vectors are used in navigation systems, such as GPS, to calculate distance and direction between two points.

Physics: Vectors are used extensively in physics to represent forces, velocities, and accelerations. For example, when a ball is thrown, its trajectory can be represented as a vector.

Computer Graphics: In computer graphics, vectors are used to represent the positions and orientations of objects in a three-dimensional space.

Engineering: Vectors are used in engineering to represent the direction and magnitude of forces acting on a structure. For example, when designing a bridge, vectors are used to calculate the load-bearing capacity of the structure.

Financial Markets: In finance, vectors are used to represent the returns of different assets in a portfolio, allowing investors to calculate the risk and return of their investments.

Overall, vectors are used in many different fields to represent physical quantities and enable calculations and predictions.
• how?
• that is how.
• What is an example of vectors being used in the real world?
• A few examples: (1) Finding the final location of an airplane by combining its speed and direction with the wind speed. Similarly, finding where a boat will end up by combining the rowing with the water current.
(2) Finding the forces pulling on nails that are used to anchor a wall-mounted sign or flagpole hanging by a chain. A more complicated problem would be designing trusses for a bridge - bridge support beams are being pushed, pulled, and bent, and the net force is determined through vector addition.
• helped a lot.
• In the section that discusses adding and subtracting vectors, it says that you can add or subtract the magnitudes to find the magnitude of the sum or difference. Doesn't this only work when the vectors move in the same direction (i.e. they are parallel)?
• As far as I can tell, you're absolutely correct. The main reason why the component form and the unit vector form of vectors were developed is so that we can add as usual since the components are always in the same direction (x components horizontal and y components vertical).

Two vectors given in magnitude and direction form must be converted to component or unit vector form before any addition/subtraction can take place.
• I just completed Quiz 3. I missed one problem and below is the problem.
Two donkeys are tied to the same pole.
One donkey pulls the pole at a strength of of 6 N.
The other donkey pulls the pole at a strength of 3.5N in a direction that is 80 degrees counterclockwise from the direction the first donkey is pulling towards.
What is the combined strength of the two donkeys pull? Which of course is the magnitude. The problem was solved using the Law of Cosines. But they used a angel of 100 degrees. How did they get an angel of 100 degrees to solve the problem when an angel of 80 degrees was given?