Finding Coplanar and Perpendicular Vectors with Given Length in 3D Vector Space

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In summary, to find the set of vectors that are coplanar with V1 and V2 and perpendicular to V3, we can use the expression V = (x,y,z) = (x',y',z') + rV1 + sV2. The vector V x V3 is perpendicular to V3, but not necessarily coplanar with V1 and V2. To find the members of this set with length 2-Root-11, we can use the dot product expression V • V3 = 0. Finally, the length of a vector is sqrt(v • v).
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Homework Statement



Given the vectors: V1=(3,1,2) and V2=(2,2,-1), find the set of vectors which are coplanar with V1 and V2 and also perpendicular to V3=(-2,1,1). Then Find the members of this set which have length 2-Root-11

Homework Equations



Dot and cross product

The Attempt at a Solution



well, a set of vectors coplanar to v1 and v2 would be V = (x,y,z) = (x',y',z') + rV1 +sV2
and a vector perpendicular to v3 in the set of vectors coplanar with v1 and v2 is: V X V3.
But what do i do with the 2-root-11?
 
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  • #2
VxV3 is not necessarily coplanar with V1 and V2. It's just perpendicular to whatever V you choose. Can you think of an expression using a dot product to express V perpendicular to V3? Finally the length of a vector is sqrt(v.v), right?
 
  • #3
okay, ill try it that way...
 

What is a 3-dimensional vector?

A 3-dimensional vector is a mathematical entity that is represented by three coordinates in space, typically denoted as (x, y, z). It is used to represent a quantity that has both direction and magnitude, such as force or velocity.

How do you add two 3-dimensional vectors?

To add two 3-dimensional vectors, you simply add the corresponding coordinates. For example, to add (1, 2, 3) and (4, 5, 6), you would get (1+4, 2+5, 3+6) which is equal to (5, 7, 9).

What is the dot product of two 3-dimensional vectors?

The dot product of two 3-dimensional vectors is a scalar value that results from multiplying the corresponding coordinates and then adding them together. It is often used to calculate the angle between two vectors or to project one vector onto another.

Can two 3-dimensional vectors be perpendicular to each other?

Yes, two 3-dimensional vectors can be perpendicular to each other if their dot product is equal to 0. This means that the vectors are at a 90 degree angle to each other.

How do you find the magnitude of a 3-dimensional vector?

To find the magnitude of a 3-dimensional vector, you can use the Pythagorean theorem. The magnitude is the square root of the sum of the squares of the three coordinates. For example, the magnitude of (3, 4, 5) would be √(3^2 + 4^2 + 5^2) = √(9 + 16 + 25) = √50 = 7.07.

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