Subtracting a value from a vector

In summary, the conversation discusses the concept of subtracting a scalar from a vector and its geometric interpretation. It is generally not meaningful to mix a scalar and vector in this operation, but if the vector happens to be a complex number, both the vector and scalar can be seen as vectors in a one-dimensional vector space. However, more information is needed to fully understand the context, such as the origin of the vector and the dimension of the vector space.
  • #1
Aristarchus_
95
7
Homework Statement
If we have a vector z, what does it mean to subtract a value of 1 from it? What is the geometric interpretation of this?
Relevant Equations
z-1
d
 
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  • #2
What are your thoughts so far? Do you think it's okay to mix a vector and a scalar in a subtraction operation like that?
 
  • #3
Aristarchus_ said:
Homework Statement:: If we have a vector z, what does it mean to subtract a value of 1 from it? What is the geometric interpretation of this?
Relevant Equations:: z-1

d
It generally makes no sense to add a scalar to a vector or to subtract a scalar from a vector. If z happens to be a complex number, then the expression ##z - 1## is treating 1 as also being a complex number (i.e., 1 + 0i), so both z and 1 are essentially vectors.
 
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  • #4
Aristarchus_ said:
Homework Statement:: If we have a vector z, what does it mean to subtract a value of 1 from it? What is the geometric interpretation of this?
Relevant Equations:: z-1

d
It means that you add two numbers, ##z## and ##1.##

Explanation: You can only add vectors from the same set*). You said that ##z## is a vector. In order to add ##1## to ##z## they have to be of the same dimension*). So ##z+1## demands to be the vector addition in a one-dimensional vector space. Now, a one-dimensional vector space is isomorphic to (~ the same as) the scalar field of the vector space, whatever this is in your case. I assume the real or complex numbers. Therefore ##z+1## is the addition of two numbers, which at the same time are vectors from a one-dimensional vector space.

You see that I already had to make several guesses to answer your question at all! What was missing?
  1. Where is ##z## from? Which vector space?
  2. What scalar field do you have? Means, if you multiply a vector by a number, where is the number from?
  3. What is the dimension of the vector space?
  4. Is it a graduated vector space that allows vectors from different dimensions?
  5. Is ##1## an abbreviation, e.g. ##1=(1,\ldots,1)##?

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*) This is not 100% true since we can have formal sums like those in a Graßmann algebra, but as you have to ask this question, I allowed myself to make the assumption of a finite-dimensional vector space of one given dimension.
 

FAQ: Subtracting a value from a vector

1. How do you subtract a value from a vector?

To subtract a value from a vector, you simply need to subtract the value from each element in the vector. For example, if you have a vector [1, 2, 3] and you want to subtract 1 from it, the resulting vector would be [0, 1, 2].

2. Can you subtract a vector from another vector?

Yes, you can subtract a vector from another vector by subtracting the corresponding elements from each other. For example, if you have two vectors [1, 2, 3] and [4, 5, 6], the resulting vector would be [-3, -3, -3].

3. What happens if the vector and value have different dimensions?

If the vector and value have different dimensions, the subtraction cannot be performed. The dimensions of both must match in order for the subtraction to be valid.

4. How does subtracting a value from a vector affect the direction of the vector?

Subtracting a value from a vector does not affect the direction of the vector. It only changes the magnitude of the vector. The direction remains the same.

5. Is subtracting a value from a vector the same as scaling the vector?

No, subtracting a value from a vector and scaling a vector are two different operations. Subtracting a value changes the magnitude of the vector, while scaling a vector changes both the magnitude and direction.

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