Solving a Vector Problem: v1 and v2 Sum to <-4,1,1> with Given Conditions

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In summary, the problem asks for finding two vectors, v1 and v2, whose sum is <-4, 1, 1>. v1 is parallel to <2, 5, -4> and v2 is perpendicular to <2, 5, -4>. To solve this problem, we can first set v1 as some constant multiple of <2, 5, -4> and then use the given information to find v2. By setting v1 as K<2, 5, -4>, we can find the coordinates of v1 and then use the equation <4, 1, 1> - K<2, 5, -4> = v2 to solve for the
  • #1
Loppyfoot
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Homework Statement



Find two vectors, v1, and v2, whose sum is <-4,1,1>, where v1 is parallel to <2,5,-4>, and where v2 is perpendicular to <2,5,-4>



Homework Equations



I am guessing I use the cross product for this equation, but I'm confused about how to start this problem.

The Attempt at a Solution



any hlep at all would be greatly appreciated!
 
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  • #2
Loppyfoot said:

Homework Statement



Find two vectors, v1, and v2, whose sum is <-4,1,1>, where v1 is parallel to <2,5,-4>, and where v2 is perpendicular to <2,5,-4>



Homework Equations



I am guessing I use the cross product for this equation, but I'm confused about how to start this problem.
Don't use the cross product.

Write v1 and v2 using coordinates. You're given that they add to <-4, 1, 1>. You're also given that v1 is parallel to <2, 5, -4>, which means that v1 is some constant multiple of <2, 5, -4>. You're also given that v2 is perpendicular to <2, 5, -4>. Two vectors being perpendicular should suggest a particular operation.
 
  • #3
So, first I realized that v1= K<2,5,-4>

Then I figured that since v1+v2=sum, ;
then:

<4,1,1> - K<2,5,-4> = v2

Now, what process would I use to find K?
 
  • #4
Loppyfoot said:
So, first I realized that v1= K<2,5,-4>
Then what are the coordinates of v1?

Loppyfoot said:
Then I figured that since v1+v2=sum, ;
then:

<4,1,1> - K<2,5,-4> = v2

Now, what process would I use to find K?
How about finding coordinates for v2 first?

Also what can you do with the given information that v2 is perpendicular to <2, 5, -4>?
 

1. What is a simple vector problem?

A simple vector problem involves using vector operations (such as addition, subtraction, and multiplication) to solve a given problem. Vectors are mathematical quantities that have both magnitude (size) and direction.

2. How do you represent a vector?

Vectors are typically represented by an arrow pointing in the direction of the vector, with the length of the arrow representing the magnitude of the vector. They can also be represented using coordinates, with the x-component and y-component indicating the direction and magnitude, respectively.

3. What are some common vector operations?

Some common vector operations include addition, subtraction, scalar multiplication, and dot product. Addition and subtraction involve combining two or more vectors, while scalar multiplication involves multiplying a vector by a scalar (a number). The dot product is a way to find the angle between two vectors or to project one vector onto another.

4. How do you solve a simple vector problem?

To solve a simple vector problem, you first need to identify the given vectors and the desired operation. Then, you can use the properties of vector operations to perform the necessary calculations. It is important to pay attention to the direction and magnitude of the vectors when solving the problem.

5. What are some real-world applications of simple vector problems?

Simple vector problems have many applications in science and engineering, such as in physics, navigation, and computer graphics. For example, vectors are used to calculate the velocity and acceleration of objects in motion, to determine the direction and strength of forces acting on an object, and to create realistic 3D graphics in video games and movies.

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