Question

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)

Answer 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).