Proving Vector Theories & Equilibrium: Exploring Questions on Zero Vectors

In summary: essentially you want to know if two perpendicular vectors add up to zero, and the theorem tells you that they do.
  • #1
Tensaiga
43
0
hello, i have a few theories questions to ask. (i don't know where to start for question such as these...)

Question: By considering the angles between the vectors, show that vector A + vector B and vector A - vector B are perpendicular when |A| = |B|.

Question: Prove for any vectors A and B, --->
that |A+B|^2 + |A-B|^2 = 2(|A|^2 +|B|^2)

Question: Three forces of 5N , 7N, 8N, are applied to an object. If the object is in a state of equillibrium, show how must the forces be arranged.

also i wonder why is zero vector's direction is undefined? is it because there is no magnitude?

Thank You
 
Last edited:
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  • #2
For your last question the three forces need to form a closed triangle. Of cause the triangle can be rotated in any direction, which means that the direction of the forces are not uniquely defined, but the angles that the vectors make with one another are fixed. Also note that by changing the order of adding the three vectors will produce two triangles that are mirror images of each other.
 
  • #3
third question: the forces must be arranged in a way that the sum of them is the zero vector.
 
  • #4
hey :)

to your first question:

[tex]
( \vec A + \vec B ) \cdot (\vec A - \vec B) = (a_1+b_1)*(a_1-b_1) + (a_2+b_2)*(a_2-b_2) + (a_3+b_3)*(a_3-b_3)\\

= a_1^2-b_1^2 + a_2^2 - b_2^2 + a_3^2 - b_3^2\\
= a_1^2+a_2^2+a_3^2 - (b_1^2+b_2^2+b_3^2)
[/tex]

if perpendicular, this is supposed to be 0, so

[tex]
a_1^2+a_2^2+a_3^2 - (b_1^2+b_2^2+b_3^2) = 0\\
a_1^2+a_2^2+a_3^2 = b_1^2+b_2^2+b_3^2\\
\sqrt{a_1^2+a_2^2+a_3^2} = \sqrt{b_1^2+b_2^2+b_3^2}\\
\Leftrightarrow |\vec A| = |\vec B|
[/tex]
 
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  • #5
sorry, there's been linebreaks missing, so here again:

hey :)

to your first question:

[tex]
( \vec A + \vec B ) \cdot (\vec A - \vec B) = (a_1+b_1)*(a_1-b_1) + (a_2+b_2)*(a_2-b_2) + (a_3+b_3)*(a_3-b_3)\\

= a_1^2-b_1^2 + a_2^2 - b_2^2 + a_3^2 - b_3^2\\
= a_1^2+a_2^2+a_3^2 - (b_1^2+b_2^2+b_3^2)
[/tex]

if perpendicular, this is supposed to be 0, so

[tex]
a_1^2+a_2^2+a_3^2 - (b_1^2+b_2^2+b_3^2) = 0\\
a_1^2+a_2^2+a_3^2 = b_1^2+b_2^2+b_3^2\\
\sqrt{a_1^2+a_2^2+a_3^2} = \sqrt{b_1^2+b_2^2+b_3^2}\\
\Leftrightarrow |\vec A| = |\vec B|
[/tex]
 
  • #6
wait, we just factor them out? wow, that's cool thanks.
But why did you mutiply the two vectors? Because they are perpendicular?

For the last question i know that their sum has to be zero, but where would you place them? why are the angles fixed? it doesn't have to fixed, it could have a degree to it, doesn't it? i know that the resultant force of two forces has to be equal to the last vector, but how...?

Thanks
 
  • #7
I multiplied them out because if I want to find something out about the angle between them, the scalar product tells you. So basically I rewrote your task to:

Proof: [tex] (\vec A + \vec B)*(\vec A - \vec B) = 0 [/tex] if [tex] |\vec A|=|\vec B| [/tex]

That's how I read your question...
 

1. What is a vector?

A vector is a mathematical quantity that has both magnitude and direction. It is represented by an arrow pointing in the direction of the vector, with the length of the arrow representing the magnitude.

2. How do you prove a vector theory?

To prove a vector theory, you need to use mathematical equations and principles to show that the theory is true for all possible cases. This can involve using vector operations such as addition, subtraction, and multiplication, as well as geometric proofs and mathematical induction.

3. What is equilibrium in terms of vectors?

In terms of vectors, equilibrium refers to a state where all forces acting on an object are balanced, resulting in a net force of zero. This means that the object will not accelerate in any direction and will remain at rest or in constant motion.

4. How do you determine if a vector is a zero vector?

A vector is considered a zero vector if it has a magnitude of zero, meaning that it has no direction or length. This can also be determined by performing vector operations, such as addition or subtraction, and if the resulting vector is also a zero vector.

5. Can a zero vector be used in vector calculations?

Yes, a zero vector can be used in vector calculations. It acts as an identity element for addition, meaning that when added to any other vector, it does not change the result. However, it cannot be used in other vector operations such as multiplication or division.

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