Finding vectors in a plane

In summary, if you have a plane with the equation x + 3y + 2z = 0, you can find two vectors in that plane by choosing values for x and y and calculating z using the equation. If x + 3y + 2z ≠ 0, the plane would look different, but you can still find two vectors in the plane using the same method.
  • #1
Cankur
9
0
Hello!

I was just thinking, say, I have a plane with the equation x + 3y + 2z = 0. How would I go about finding two vectors in that plane (that are not parallell)?

Sidenote: I pretty much get the intuition of how a plane with the given equation would look but what would happen if x + 3y + 2z ≠ 0? How would that plane look?

Thanks in advance!
 
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  • #2
pick values for x and y, and use your equation to calculate z:
If x= 1, y= 0, then 1+ 3(0)+ 2z= 0 so z= -1/2. A point in that plane is (1, 0, -1/2) so the vector from (0, 0, 0) to (1, 0, -1/2), i- (1/2)k, is a vector in that plane.

If x= 0, y= 1, then 0+ 3(1)+ 2z= 0 so z= -3/2. A point in that plane is (0, 1, -3/2) so the vector from (0, 0, 0) to (0, 1, -3/2), j-(3/2)k, is another vector in that plane.

Of course, since those two vectors are independent, they span the plane.
 

1. What is a vector in a plane?

A vector in a plane is a mathematical object that has both magnitude (length) and direction. It is usually represented by an arrow pointing in the direction of the vector with a specific length to represent its magnitude.

2. How do you find the magnitude of a vector in a plane?

The magnitude of a vector in a plane can be found using the Pythagorean theorem. This involves finding the length of the vector by taking the square root of the sum of the squares of its components (x and y).

3. How do you find the direction of a vector in a plane?

The direction of a vector in a plane can be found using trigonometric functions such as sine, cosine, and tangent. The direction is typically measured in degrees or radians from a reference axis.

4. What is the difference between a position vector and a displacement vector?

A position vector specifies the location of a point in a plane relative to a fixed origin, while a displacement vector represents the change in position of an object from its original location to its final location. In other words, a displacement vector is the difference between two position vectors.

5. How do you add or subtract vectors in a plane?

Vectors in a plane can be added or subtracted by adding or subtracting their respective components (x and y). This can be visualized using the triangle method, where the two vectors are placed end to end and the resulting vector is drawn from the first vector's tail to the second vector's head.

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