Test Question: Vector Proof Help

In summary, the conversation discusses whether there exists a vector in R3 such that v*v = ||v||, and the person realizes that it is true when considering an arbitrary unit vector. They also address any potential objections from classmates and conclude that the equation holds true for any unit vector.
  • #1
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Sorry for notation guys, but I don't know how to use LaTex.

Homework Statement


True or false, there exists a vector v (or set of vectors) in R3 such that v*v = ||v|| (v dot v equals the magnitude of v).


Homework Equations


At first I thought this was false, but then I considered an arbitrary unit vector v=(1/(14)/\(-1/2)*(1,2,3)...in words: one over the square root of fourteen times the vector one, two, three (the unit vector of (1,2,3)).


The Attempt at a Solution



Taking v*v, u get (1*1)/14 + (2*2)/14 + (3*3)/14 = 1/14 + 4/14 + 9/14 = 14/14 = 1, which is trivial for a unit vector. Also, the magnitude is just the square root of this answer, since the components are already squared for dotting itself, which is one; again, trivial. Am I correct or can I not consider a unit vector?
 
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  • #2
You just demonstrated that the answer it true. Why are you doubting yourself?
 
  • #3
I am doubting myself because I had strong opposition from 3 classmates that for some reason you can't use a unit vector for the proof.
 
  • #4
Well, we can't really tell you if it said that you can't use unit vectors. Regardless, it's clearly true for any unit vector because, using your notation, the equation says ||v||^2 = v*v = ||v||.

Edit: Yeah, I'm really out of it today.
 
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What is a vector and its properties?

A vector is a quantity that has both magnitude and direction. It can be represented by an arrow, with the length of the arrow representing the magnitude and the direction of the arrow indicating the direction. The properties of a vector include its magnitude, direction, and its components (x and y coordinates).

What is a vector proof and why is it important?

A vector proof is a mathematical demonstration that shows the relationship between different vectors using various mathematical operations and properties. It is important because it allows us to understand and analyze the behavior of vectors in different situations, and use them to solve real-world problems in physics, engineering, and other fields.

How do you perform vector addition and subtraction?

Vector addition is performed by adding the corresponding components of two or more vectors. For example, the sum of two vectors A(x1, y1) and B(x2, y2) is given by A + B = (x1 + x2, y1 + y2). Vector subtraction is similar, but with the difference of the corresponding components. For example, the difference between two vectors A(x1, y1) and B(x2, y2) is given by A - B = (x1 - x2, y1 - y2).

What is a dot product and how is it calculated?

A dot product is a mathematical operation that takes two vectors and produces a scalar (a single numerical value) as the result. It is calculated by multiplying the corresponding components of two vectors and adding the products. For example, the dot product of two vectors A(x1, y1) and B(x2, y2) is given by A · B = x1x2 + y1y2.

What is a cross product and how is it calculated?

A cross product is a mathematical operation that takes two vectors and produces a vector as the result. It is calculated by using the following formula: A × B = (|A| * |B| * sinθ) * n, where θ is the angle between the two vectors and n is a unit vector perpendicular to both A and B. This means that the result of a cross product is a vector that is perpendicular to both A and B.

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