Asked by Ayansa Dabesa soboka
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)
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)
Answers
Answered by
GPT 3.5
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).
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).
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