Simple Vector Proof: sa+ta=(s+t)a, s*(ta)=(s*t)a

In summary, we are trying to show that sa+ta=(s+t)a and s*(ta)=(s*t)a. To prove this, we must carefully check the axioms given for our vector space. Additionally, we can show this geometrically by drawing the vectors sa and ta, which are parallel to a, and adding them together to get the resultant vector of (s+t)a. This is also true for s*(ta), which is equal to (s*t)a. Therefore, the two equations are proven through both algebraic and geometric methods.
  • #1
grimster
39
0
a is a vector and s and t are two integers. I'm supposed to show that:

sa+ta=(s+t)a

and

s*(ta)=(s*t)a

the two are so obvious I'm not sure how i prove them.
 
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  • #2
Make a careful check of the axioms given for your vector space, and see what needs to be proven.
 
  • #3
thing is, i think I'm supposed to show it geometrically by drawing it. how do i do that?
 
  • #4
let a = (x,y) in the component form
then
sa+ta =
s(x,y) + t(x,y) =
(sx,sy) + (tx,ty)=
(sx+tx,sy+ty)=

(x(s+t),y(s+t))=

(s+t)(x,y)=
(s+t)a and do the similar for b)
 
  • #5
Well, sa is parallell to a, isn't it?
And so is ta..
So, how would you geometrically add these vectors, and what resultant vector does this equal?
 
  • #6
arildno said:
Well, sa is parallell to a, isn't it?
And so is ta..
So, how would you geometrically add these vectors, and what resultant vector does this equal?

(s+t)*a

but that is what I'm supposed to show. so is it enough to just draw sa and then ta from where sa ends? add them together so to speak?
 
  • #7
I guess so.
 
  • #8
That's about as geometrically proven as it gets...
 

Related to Simple Vector Proof: sa+ta=(s+t)a, s*(ta)=(s*t)a

1. What is a simple vector proof?

A simple vector proof is a mathematical demonstration that shows how a particular vector equation or identity is true. It involves using basic vector operations and properties to logically derive the given equation.

2. How do you prove sa+ta=(s+t)a using vectors?

To prove sa+ta=(s+t)a, we can use the distributive property of vector multiplication. First, we expand the left side to (s+t)a, then we apply the distributive property to get sa+ta. This shows that both sides are equal and the equation is true.

3. Can you provide an example of a simple vector proof?

One example of a simple vector proof is proving the associative property of vector addition, (a+b)+c = a+(b+c). We can do this by expanding both sides and showing that the components are equal, thus proving the identity to be true.

4. What is the difference between scalar and vector multiplication?

Scalar multiplication involves multiplying a vector by a single number or scalar, while vector multiplication involves multiplying two vectors together. Scalar multiplication results in a scalar quantity, while vector multiplication results in a vector quantity.

5. How are vector proofs used in science?

Vector proofs are used in a variety of scientific fields, such as physics, engineering, and computer graphics. They help scientists and researchers understand and manipulate vector quantities in mathematical models and equations, which are often used to describe physical phenomena and solve real-world problems.

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