Squaring of vectors in absolute value

In summary, the squared vector sum of |\vec{a}+\vec{b}|^{2} and (\vec{a}+\vec{b})^{2} are equal, assuming the implied multiplication is the dot product. The result is also equal to \sqrt{a^2+b^2}, which represents the magnitude of the two vectors. This is only true if the vectors are orthogonal.
  • #1
M. next
382
0
Is |[itex]\vec{a}[/itex]+[itex]\vec{b}[/itex]|[itex]^{2}[/itex] equal to the same thing as ([itex]\vec{a}[/itex]+[itex]\vec{b}[/itex])[itex]^{2}[/itex]? And when is it equal to √(a[itex]^{2}[/itex]+b[itex]^{2}[/itex])?

Thanks.
 
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  • #2
M. next said:
Is |[itex]\vec{a}[/itex]+[itex]\vec{b}[/itex]|[itex]^{2}[/itex] equal to the same thing as ([itex]\vec{a}[/itex]+[itex]\vec{b}[/itex])[itex]^{2}[/itex]?
They're the same, assuming the implied multiplication in the expression on the right is the dot product. Otherwise, multiplication of one vector by another is not defined (with the exception of the cross product).
M. next said:
And when is it equal to √(a[itex]^{2}[/itex]+b[itex]^{2}[/itex])?

Tip: You don't need so many tex or itex tags. Your squared vector sum can be written like this:
[itex](\vec{a} +\vec{b})^2 [/itex]
Or instead of the itex tags, you can use ## delimiters at the front and back.
 
  • #3
Thank you for the reply and the tip. But about the √(a^2+b^2)?? My second part of the question?
 
  • #4
M. next said:
Thank you for the reply and the tip. But about the √(a^2+b^2)?? My second part of the question?

That is merely the magnitude of both vectors. Assuming that's what you mean? You were a little unclear on the second part. Think of magnitude as the size or length of those two vectors.
 
  • #5
M. next said:
Thank you for the reply and the tip. But about the √(a^2+b^2)?? My second part of the question?

[itex]\sqrt{a^2+b^2}[/itex] is the magnitude of [itex]\vec{a}±\vec{b} [/itex],where [itex]\vec{a}[/itex] and [itex]\vec{b}[/itex] are orthogonal (perpendicular) vectors.
 
  • #6
Okay. Thank you, yes, it is exactly what I meant.
 

What is the definition of squaring a vector in absolute value?

Squaring a vector in absolute value involves taking each component of the vector and squaring it, then adding all of the squared components together and taking the square root of the sum. This results in a single number, the absolute value of the vector.

How is squaring a vector in absolute value different from finding the magnitude of a vector?

Squaring a vector in absolute value is similar to finding the magnitude of a vector, as both involve taking the square root of the sum of squared components. However, squaring in absolute value also involves taking the absolute value of each component before squaring, while finding the magnitude does not.

Why is squaring a vector in absolute value useful?

Squaring a vector in absolute value is useful because it allows us to find the magnitude or length of a vector, which can be important in many applications such as physics, engineering, and mathematics. It also allows us to compare the magnitudes of different vectors.

Can a vector have a negative absolute value after squaring?

No, a vector cannot have a negative absolute value after squaring. The absolute value function always returns a non-negative value, so even if the vector has negative components, squaring and then taking the absolute value will result in a positive value.

Are there any other methods for finding the absolute value of a vector?

Yes, there are other methods for finding the absolute value of a vector, such as using the Pythagorean theorem or using the dot product of the vector with itself. However, squaring and taking the square root is the most common and efficient method.

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