a) Find all vectors that are orthogonal to E1 = (1, 0, 0)
b) Find all vectors that are orthogonal to both E1 and E3 = (0, 0, 1)
c) Find all vectors that are orthogonal to E1, E2 and E3 = (0, 0, 1)
a) To find all vectors orthogonal to E1 = (1, 0, 0), we need to find all vectors (x, y, z) such that (x, y, z) · (1, 0, 0) = 0, where · represents the dot product.
The dot product between two vectors can be calculated by multiplying corresponding components and summing them, so we have:
x*1 + y*0 + z*0 = 0
x = 0
Thus, any vector of the form (0, y, z) where y and z can be any real numbers is orthogonal to E1.
b) To find all vectors orthogonal to both E1 = (1, 0, 0) and E3 = (0, 0, 1), we need to find all vectors (x, y, z) such that:
(x, y, z) · (1, 0, 0) = 0
(x, y, z) · (0, 0, 1) = 0
Using the dot product, we can write these equations:
x*1 + y*0 + z*0 = 0
0 + 0 + z*1 = 0
Simplifying, we have:
x = 0
z = 0
Thus, any vector of the form (0, y, 0) where y can be any real number is orthogonal to both E1 and E3.
c) To find all vectors orthogonal to E1 = (1, 0, 0), E2 = (0, 1, 0), and E3 = (0, 0, 1), we need to find all vectors (x, y, z) such that:
(x, y, z) · (1, 0, 0) = 0
(x, y, z) · (0, 1, 0) = 0
(x, y, z) · (0, 0, 1) = 0
Using the dot product, we can write these equations:
x*1 + y*0 + z*0 = 0
x*0 + y*1 + z*0 = 0
x*0 + y*0 + z*1 = 0
Simplifying, we have:
x = 0
y = 0
z = 0
Thus, the only vector that is orthogonal to E1, E2, and E3 is the zero vector (0, 0, 0).
a) To find vectors that are orthogonal to E1 = (1, 0, 0), we need to find vectors that have a dot product of zero with E1.
Let's assume the vector we want to find is (x, y, z).
The dot product of (x, y, z) and E1 = (1, 0, 0) is given by:
x*1 + y*0 + z*0 = x
For the dot product to be zero, x must be zero.
Therefore, any vector of the form (0, y, z), where y and z can be any real numbers, is orthogonal to E1.
b) To find vectors that are orthogonal to both E1 = (1, 0, 0) and E3 = (0, 0, 1), we need to find vectors that have a dot product of zero with both E1 and E3.
Let's assume the vector we want to find is (x, y, z).
The dot product of (x, y, z) and E1 = (1, 0, 0) is given by:
x*1 + y*0 + z*0 = x
The dot product of (x, y, z) and E3 = (0, 0, 1) is given by:
x*0 + y*0 + z*1 = z
For the dot product to be zero with both E1 and E3, x must be zero and z must be zero.
Therefore, any vector of the form (0, y, 0), where y can be any real number, is orthogonal to both E1 and E3.
c) To find vectors that are orthogonal to E1 = (1, 0, 0), E2 = (0, 1, 0), and E3 = (0, 0, 1), we need to find vectors that have a dot product of zero with all three of these vectors.
Let's assume the vector we want to find is (x, y, z).
The dot product of (x, y, z) and E1 = (1, 0, 0) is given by:
x*1 + y*0 + z*0 = x
The dot product of (x, y, z) and E2 = (0, 1, 0) is given by:
x*0 + y*1 + z*0 = y
The dot product of (x, y, z) and E3 = (0, 0, 1) is given by:
x*0 + y*0 + z*1 = z
For the dot product to be zero with all three vectors, x must be zero, y must be zero, and z must be zero.
Therefore, the zero vector (0, 0, 0) is orthogonal to E1, E2, and E3.