To find two unit vectors that are orthogonal to both vectors v1 = (7, 5, 1) and v2 = (1, 1, 0), we first need to determine a vector that is perpendicular to both. This can be accomplished using the cross product.
First, we calculate the cross product of the two vectors:
v1 x v2 =
(i, j, k)
(7, 5, 1)
(1, 1, 0)
Using the determinant method, we expand:
= i(5*0 – 1*1) – j(7*0 – 1*1) + k(7*1 – 5*1)
= i(0 – 1) – j(0 – 1) + k(7 – 5)
= -i + j + 2k
This gives us the vector v3 = (-1, 1, 2).
Next, we need to convert this vector into a unit vector. To do that, we calculate its magnitude:
Magnitude of v3 = ||v3|| = √((-1)² + (1)² + (2)²) = √(1 + 1 + 4) = √6.
Now we can find the unit vector by dividing each component of v3 by its magnitude:
Unit vector u1 = v3 / ||v3|| = (-1/√6, 1/√6, 2/√6) = (-√6/6, √6/6, 2√6/6).
Next, we can find another unit vector that is also orthogonal to both vectors. This can be done by taking the negative of u1:
u2 = (√6/6, -√6/6, -2√6/6).
So, the two unit vectors orthogonal to both (7, 5, 1) and (1, 1, 0) are:
- u1 = (-√6/6, √6/6, 2√6/6)
- u2 = (√6/6, -√6/6, -2√6/6)