Question

(15 points) Find an orthonormal basis of the plane x1 + 4x2 – x3 = 0.

Answer 1

To find an orthonormal basis of the plane x1 + 4x2 – x3 = 0, we can choose two vectors that satisfy this equation and then normalize them.

Step-by-step explanation:

To find an orthonormal basis of the plane x1 + 4x2 – x3 = 0, we first need to find a basis for the plane. The equation x1 + 4x2 – x3 = 0 can be rewritten as a linear combination of the variables: x1 = -4x2 + x3. We can choose two vectors that satisfy this equation, for example:

v1 = [-4, 1, 0]

v2 = [0, 0, 1]

Next, we normalize these vectors by dividing each vector by its Euclidean norm. The Euclidean norm of a vector [a, b, c] is given by the formula sqrt(a^2 + b^2 + c^2). The normalized vectors are:

n1 = v1 / sqrt(v1[0]^2 + v1[1]^2 + v1[2]^2)

n2 = v2 / sqrt(v2[0]^2 + v2[1]^2 + v2[2]^2)

Therefore, an orthonormal basis for the plane x1 + 4x2 – x3 = 0 is:

n1 = [-0.894, 0.224, 0]

n2 = [0, 0, 1]

Answer 2

To find an orthonormal basis of the given plane, we can find two vectors that lie in the plane and are orthogonal to each other. Then, normalize these vectors to obtain the orthonormal basis. In this case, the orthonormal basis of the plane x1 + 4x2 – x3 = 0 is {u₁, u₂}, where u₁ = (1/√17, 0, 4/√17) and u₂ = (4/√18, 1/√18, 1/√18).

Step-by-step explanation:

To find an orthonormal basis of the plane x1 + 4x2 – x3 = 0, we need to find two orthogonal vectors that lie in this plane and then normalize them to obtain a unit length. Let's start by finding a vector that satisfies the equation. We can choose v₁ = (1, 0, 4) as one vector in the plane. Now, let's find a vector that is orthogonal to v₁ and lies in the plane. We can choose the normal vector to the plane as v₂ = (4, 1, 1). Next, we need to normalize these vectors to obtain the orthonormal basis. Normalizing v₁, we get u₁ = (1/√17, 0, 4/√17), and normalizing v₂, we get u₂ = (4/√18, 1/√18, 1/√18). Therefore, the orthonormal basis of the plane is {u₁, u₂}.