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

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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.
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AkilMAI
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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
 
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  • #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).

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