Given a vector u, the familiar absolute-value notation |u| is often used for its magnitude.
Thus the expressions u•u and |u|2 both mean the same thing. What exactly do they mean?
let u = [a,b]
then u•u = a^2 + b^2
also |u| = √(a^2 + b^2)
then |u|^2 = a^2 + b^2
A geometric interpretation would be the area of a square whose sides equals the magnitude of the vector.
e.g. let u = [3,4]
then |u| = √(9+16) = 5
so the magnitude of the vector is 5
a square with sides 5 would have an area of 25
and |u|^2 = 25
Don't know if this is what you are after.
Ah, the age-old question of vector magnitudes. Well, my dear inquirer, brace yourself for some mathematical hilarity!
When we say u•u, we're referring to the dot product of vector u with itself. It's like asking u to do a little dance and then multiply its components together. The result is a single number that represents the magnitude squared of u.
Now, onto |u|2! This notation simply means taking the absolute value of vector u and then squaring it. It's like giving u a big, warm hug and then raising it to the power of 2. The outcome is also the magnitude squared of u.
So, whether you prefer the dot product or the absolute value squared, both expressions paint a picture of the magnitude of vector u. Think of it as looking at u from different directions, but ultimately ending up at the same humorous punchline. Magnitude squared, squared magnitude - it's just a comical way to represent the same concept.
The expression u•u represents the dot product of vector u with itself. In other words, it is the sum of the products of the corresponding components of u. Mathematically, u•u = u₁u₁ + u₂u₂ + ... + uₙuₙ.
On the other hand, |u|² represents the square of the magnitude of vector u. The magnitude of a vector is the length or absolute value of the vector. In the case of a vector with n components, the magnitude is calculated using the Pythagorean theorem, which involves taking the square root of the sum of the squares of the components. In this case, the magnitude squared is obtained by squaring each component of u and summing them up. Mathematically, |u|² = u₁² + u₂² + ... + uₙ².
Therefore, both u•u and |u|² represent the same thing, which is the squared magnitude of vector u.
The notation u•u represents the dot product of vector u with itself. The dot product is a mathematical operation that takes two vectors and produces a scalar quantity. In this case, u•u represents the dot product of vector u with itself, which results in the magnitude (or length) of vector u squared.
On the other hand, |u|2 represents the magnitude of vector u squared. The magnitude of a vector is a measure of its length or size. By squaring the magnitude, we eliminate the possibility of negative values and obtain a positive scalar.
Both u•u and |u|2 represent the square of the magnitude of vector u. This value can provide important information about the vector, such as its length or its similarity to other vectors.