What is a Non-zero Vector in R^3 that Belongs to Two Given Spans?

  • Thread starter AkilMAI
  • Start date
  • Tags
    Vector
In summary, the conversation discusses finding a non-zero vector in R^3 that belongs to two different spans. The solution involves using the scalar triple product and determining the vector's position on the intersection of two planes. The final solution is x=1, y=0, and z=4.
  • #1
AkilMAI
77
0

Homework Statement



I require some help to find a non-zero vector in R^3 that belongs both to span {y; u} and to span {v;w} where y = (1; 0; 0); u = (0; 0; 1), v = (1; 1; 1) and w = (2; 3;-1),
I need to know if my below solutions is ok.Thank you

2. The attempt at a solution
Let 'a' be the required vector.
I need to satisfy [a,y,u]=0 and [a,v,w]=0; where [a,y,u] is the scalar triple product of a, y and u.
Since span of two given vectors is a plane, 'a' lies on the intersection of two planes hence 'a' is the vector along the line of intersection of the two planes.
=>...[A,B,C]=det(ABC)...so det(a,y,u)=>y=0...and det(a,v,w)=-4x+y+z=0...y=0,so x=1 and z=4
 
Physics news on Phys.org
  • #2
Right!
 
  • #3
ok:)...thanks
 

FAQ: What is a Non-zero Vector in R^3 that Belongs to Two Given Spans?

1. What is a non-zero vector in R^3?

A non-zero vector in R^3 is a three-dimensional vector with at least one non-zero component. It is represented by an ordered triple (x, y, z) where x, y, and z are real numbers.

2. How is a non-zero vector represented in R^3?

A non-zero vector in R^3 can be represented as an arrow in three-dimensional space with its tail at the origin and its head at the point (x, y, z). The length of the arrow represents the magnitude of the vector and the direction of the arrow represents the direction of the vector.

3. What is the magnitude of a non-zero vector in R^3?

The magnitude, or length, of a non-zero vector in R^3 is calculated using the Pythagorean theorem. It is the square root of the sum of the squares of its components, or √(x^2 + y^2 + z^2).

4. How is the direction of a non-zero vector in R^3 determined?

The direction of a non-zero vector in R^3 is determined by the angles it makes with the positive x, y, and z axes. These angles can be found using trigonometric functions such as sine, cosine, and tangent.

5. What is the difference between a zero vector and a non-zero vector in R^3?

A zero vector in R^3 has all of its components equal to 0, while a non-zero vector has at least one non-zero component. This means that a non-zero vector has a magnitude and direction, while a zero vector has no magnitude or direction and is essentially a point at the origin (0, 0, 0).

Similar threads

Given specific v, dimension of subspace of L(V, W) where Tv=0?
Replies
15
Views
1K
Given subspaces ##U \& W##, show they are equal | Linear Algebra
Replies
8
Views
1K
Finding a complementary subspace ##U## | Linear Algebra
Replies
4
Views
2K
Find the equation of the tangent plane and normal to a surface
Replies
1
Views
651
Unable to simplify dS (Stokes' theorem)
Replies
1
Views
845
Flux of a vector field through a surface
Replies
8
Views
2K
Direction in which directional derivative is zero
Replies
6
Views
1K
Obtain an equation of the plane in the form ##px+qy+rz=d##
Replies
5
Views
598
[Linear Algebra] Show that H ∩ K is a subspace of V
Replies
5
Views
6K
Proof of a vector identity in electromagnetism
Replies
9
Views
1K

Hot Threads

Recent Insights

Back
Top