Final answer:
To find the two unit vectors orthogonal to both j - k and i + j, we use the cross product method, resulting in î - â and its negation -î + â, with the latter having the smaller i-component.
Step-by-step explanation:
To find two unit vectors orthogonal to both j - k and i + j, we can use the cross product method. These two given vectors lie in the y-z and x-y planes, respectively, and their cross product will be orthogonal to both.
To perform the cross product, we use the determinant of a 3x3 matrix where the first row contains the unit vectors î, â, and îk, the second row contains the components of the first vector, and the third row contains the components of the second. The unit vectors need to follow a cyclic order to ensure the result is a unit vector. According to the corkscrew rule and their cyclic order, the cross product of these vectors would yield a third vector orthogonal to both.
The calculation goes as follows:
î(â • k - 0 • 1) - â(î • k - 0 • 1) + îk(î • 1 - â • 1) = î(1) - â(1) + îk(0) = î - â.
This gives us one of the orthogonal unit vectors, which is î - â. Note that since both vectors need to be orthogonal to the original two vectors, and we're looking for unit vectors, the other vector can be found by simply negating all the components of the found vector (since both will have the same length and will be orthogonal to the original vectors), giving us -î + â as the other vector.
Comparing the i-components, we can see that -î + â has the smaller i-component than î - â.