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 a vector that is orthogonal to E1 = (1, 0, 0), we can choose any vector that has a dot product of zero with E1. Let's call this vector V = (a, b, c).
The dot product between E1 and V is: (1, 0, 0)·(a, b, c) = 1·a + 0·b + 0·c = a.
So, we need a vector V = (a, b, c) where a = 0. This means that the vector V must have its first component equal to zero.
Therefore, the vectors orthogonal to E1 = (1, 0, 0) are of the form V = (0, b, c), where b and c can take any real values.
b) To find a vector that is orthogonal to both E1 = (1, 0, 0) and E3 = (0, 0, 1), we can find the cross product of E1 and E3.
The cross product of two vectors is given by:
E1 × E3 = (0, 0, 1) × (1, 0, 0) = (0·0 - 1·0, 1·0 - 0·0, 0·0 - 0·1) = (0, 0, 0).
This means that the cross product of E1 and E3 is the zero vector. Therefore, there are no vectors that are orthogonal to both E1 and E3.
c) To find a vector that is orthogonal to E1 = (1, 0, 0), E2 = (0, 1, 0), and E3 = (0, 0, 1), we can find the cross product of E1 and E2.
The cross product of two vectors is given by:
E1 × E2 = (1, 0, 0) × (0, 1, 0) = (0·0 - 0·1, 0·0 - 1·0, 1·1 - 0·0) = (0, 0, 1).
Therefore, the vector (0, 0, 1) is orthogonal to E1, E2, and E3.
However, this vector is already one of the given vectors in the problem statement, so we need to find additional vectors that are orthogonal to E1, E2, and E3.
One vector that is orthogonal to all three vectors is the zero vector (0, 0, 0).
Thus, the vectors that are orthogonal to E1, E2, and E3 are any linear combination of (0, 0, 0) and (0, 0, 1).
In other words, the vectors of the form V = (0, 0, c), where c can take any real value, are orthogonal to E1, E2, and E3.
a) To find all 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 consider a vector V = (x, y, z) that is orthogonal to E1. The dot product between V and E1 is given by:
V · E1 = x * 1 + y * 0 + z * 0 = x
For the dot product to be zero, we must have x = 0. Therefore, any vector of the form (0, y, z) is orthogonal to E1, where y and z can be any real numbers.
b) To find all 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 consider a vector V = (x, y, z) that is orthogonal to E1 and E3. The dot products between V and E1, and V and E3 are given by:
V · E1 = x * 1 + y * 0 + z * 0 = x
V · E3 = x * 0 + y * 0 + z * 1 = z
For the dot products to be zero, we must have x = 0 and z = 0. Therefore, any vector of the form (0, y, 0) is orthogonal to both E1 and E3, where y can be any real number.
c) To find all 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 vectors.
Let's consider a vector V = (x, y, z) that is orthogonal to E1, E2, and E3. The dot products between V and E1, V and E2, and V and E3 are given by:
V · E1 = x * 1 + y * 0 + z * 0 = x
V · E2 = x * 0 + y * 1 + z * 0 = y
V · E3 = x * 0 + y * 0 + z * 1 = z
For the dot products to be zero, we must have x = 0, y = 0, and z = 0. Therefore, the only vector that is orthogonal to E1, E2, and E3 is the zero vector (0, 0, 0).