Final answer:
To find two unit vectors orthogonal to given vectors a and b, calculate the cross product of a and b to get a vector orthogonal to both. Normalize this vector to turn it into a unit vector. For the second unit vector, take the negative of the first.
Step-by-step explanation:
The question involves finding two unit vectors that are orthogonal to two given vectors. Orthogonal vectors are vectors that are perpendicular to each other, meaning their dot product is zero. The cross product of the original vectors gives a vector that is orthogonal to both. In this case, to find such a vector perpendicular to vectors a and b, we compute the cross product a × b. Since the vectors are three-dimensional, the cross product is also a three-dimensional vector, which can be calculated using the determinant of a matrix:
- The matrix's first row consists of the unit vectors i, j, and k.
- The second row contains the components of vector a.
- The third row contains the components of vector b.
The cross product of a = ⟨4,5,0⟩ and b = ⟨0,1,-3⟩ can be calculated as follows:
i(5*(-3) - 0*1) - j(4*(-3) - 0*0) + k(4*1 - 5*0) = ⟨0, 12, 4⟩
This gives us a non-unit vector that is orthogonal to both a and b. We then normalize this vector to find the unit vector by dividing by its magnitude, yielding the first unit vector.
Since we need two unit vectors orthogonal to a and b, and they are not parallel, their cross product will give only one vector that is perpendicular to both. To find a second vector, we can simply take the negative of the first unit vector because if a vector is orthogonal to other vectors, so is its negative.
The first unit vector orthogonal to both a and b is ⟨0, 12/√(12² + 4²), 4/√(12² + 4²)⟩, and the second vector is the negative of the first, ⟨0, -12/√(12² + 4²), -4/√(12² + 4²)⟩.