# Condition to three vectors being collinear

• LCSphysicist
In summary: In other words, they lie on a single straight line. The condition they are referring to is simply the fact that the vectors are collinear, not any specific mathematical condition.
LCSphysicist
Homework Statement
Find a condition, just using operation of vector space, such that the vectors u,v,w belong to the subspace E be colinear
Relevant Equations
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Now i am rather confused, the answer apparently is that ##(w-u) = \lambda(u-v)##

But, i could find a way that disprove the answer, that is:
Be u v and w vectors belong to R2, a subspace of R3:

What do you think? This is rather strange.

LCSphysicist said:
Homework Statement:: Find a condition, just using operation of vector space, such that the vectors u,v,w belong to the subspace E be colinear
Relevant Equations:: \n

Now i am rather confused, the answer apparently is that ##(w-u) = \lambda(u-v)##
What this is saying is that ##\vec w - \vec u## is a scalar multiple of ##\vec u - \vec v##. In other words, the two vector differences point in the same or opposite directions.
LCSphysicist said:
But, i could find a way that disprove the answer, that is:
Be u v and w vectors belong to R2, a subspace of R3:

View attachment 272763

What do you think? This is rather strange.
Your example is not a counterexample: it does not disproved the book's solution.
From your drawing ##\vec w - \vec u## can be drawn from the common point (the point where all three vectors start), and pointing straight up. ##\vec u - \vec v## can be drawn also from the common point and pointing straight up.
To draw ##\vec w - \vec u##, that's really the same as ##\vec w + (-1)\vec u##, so go out to the end of ##vec w## and then go backwards the length of ##\vec u. That should take you to a point directly above the common point.

BTW, in English we don't say "Be u v and w vectors" -- we say "Let u, v, and w be vectors that belong to ..."

member 587159 and LCSphysicist
Mark44 said:
What this is saying is that ##\vec w - \vec u## is a scalar multiple of ##\vec u - \vec v##. In other words, the two vector differences point in the same or opposite directions.

Your example is not a counterexample: it does not disproved the book's solution.
From your drawing ##\vec w - \vec u## can be drawn from the common point (the point where all three vectors start), and pointing straight up. ##\vec u - \vec v## can be drawn also from the common point and pointing straight up.
To draw ##\vec w - \vec u##, that's really the same as ##\vec w + (-1)\vec u##, so go out to the end of ##vec w## and then go backwards the length of ##\vec u. That should take you to a point directly above the common point.

BTW, in English we don't say "Be u v and w vectors" -- we say "Let u, v, and w be vectors that belong to ..."
Yes, reading again with your information makes me realize that the question is not saying that ##\vec w - \vec u## is a scalar multiple of ##\vec u - \vec v## IMPLIES the collinear condition, but it is saying that if the vectors are collinear, so this is true.
While i was using "iff", the book was using the right "if"

"BTW, in English we don't say "Be u v and w vectors" -- we say "Let u, v, and w be vectors that belong to ..."
Thank you for this clarification.

LCSphysicist said:
the question is not saying that w→−u→ is a scalar multiple of u→−v→ IMPLIES the collinear condition
It is saying that.
"a condition such that [if satisfied by the vectors then] the vectors u,v,w [are] colinear"
You may be confused by the way collinearity is being used here. They mean that the points represented by the vectors are collinear, as in your diagram.

## 1. What does it mean for three vectors to be collinear?

Collinear vectors are three or more vectors that lie on the same line or are parallel to each other. This means that they have the same direction and can be expressed as scalar multiples of each other.

## 2. How can I determine if three vectors are collinear?

To determine if three vectors are collinear, you can use the cross product or dot product. If the cross product of two of the vectors is equal to the third vector, then they are collinear. Also, if the dot product of any two of the vectors is equal to the product of their magnitudes, then they are collinear.

## 3. Can three non-zero vectors be collinear?

No, three non-zero vectors cannot be collinear. Collinear vectors must have the same direction, which is not possible with three different non-zero vectors.

## 4. What is the significance of three vectors being collinear?

When three vectors are collinear, it means that they are linearly dependent. This can be useful in solving systems of linear equations or in understanding the relationships between different vectors.

## 5. Can three vectors in three-dimensional space be collinear?

Yes, three vectors in three-dimensional space can be collinear. As long as they lie on the same line or are parallel to each other, they are considered collinear.

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