Question

Find a vector orthogonal to both ⟨−5,1,0⟩ and to ⟨0,1,5⟩

Answer 1

To find a vector orthogonal to both ⟨−5,1,0⟩ and ⟨0,1,5⟩, calculate their cross product. The resulted vector ⟨5, 25, −1⟩ is orthogonal to both given vectors.

Step-by-step explanation:

To find a vector that is orthogonal to both ⟨−5,1,0⟩ and ⟨0,1,5⟩, we can take the cross product of these two vectors. The cross product of two vectors results in a vector that is perpendicular to both of the original vectors. The cross product is defined by the determinant of a matrix formed by the unit vectors î, î, and  along with the components of the vectors in question.

Let's calculate this step by step:

1. Write the vectors in component form: A = ⟨−5, 1, 0⟩ and B = ⟨0, 1, 5⟩.
2. Set up the determinant with the unit vectors in the first row:
   
3. Calculate the determinant, which gives us the orthogonal vector: C = A × B = ⟨(1⋅ 5 − 0⋅ 1), (−5⋅ 5 − 0⋅ −0), (−5⋅ 1 − 1⋅ 0)⟩ = ⟨5, 25, −1⟩.

The resulting vector C = ⟨5, 25, −1⟩ is orthogonal to both A and B.