Determine two unit vectors that are perpendicular to both a=[-5,6,2] and d=[3,3,-8].

1. find the cross product of the two given vectors, that will be a perpendicular

2. divide that vector by its magnitude to make it a unit vector

3. switch the signs to get the other unit vector.

To find unit vectors that are perpendicular to both given vectors, we can use the cross product of the two vectors.

The cross product of two vectors is a vector that is orthogonal (perpendicular) to both of them. Here's how you can find it:

1. Let's calculate the cross product of the given vectors:

a x d = [(-5) * (-8) - 6 * 3, (-5) * 3 - 2 * (-8), 6 * 3 - 2 * 3]
= [16, -7, 12]

2. Now, we have the cross product vector [16, -7, 12]. To find unit vectors which are perpendicular to both a and d, we need to divide this vector by its magnitude.

Magnitude of the cross product vector = sqrt(16^2 + (-7)^2 + 12^2) = sqrt(449) ≈ 21.18

3. Divide the cross product vector by its magnitude to obtain the unit vectors:

v1 = [16/21.18, -7/21.18, 12/21.18] ≈ [0.756, -0.331, 0.567]
v2 = -v1 ≈ [-0.756, 0.331, -0.567]

Note: We can also obtain a second vector by negating v1 since it will still be perpendicular to both a and d.

Therefore, the two unit vectors that are perpendicular to both a=[-5,6,2] and d=[3,3,-8] are approximately [0.756, -0.331, 0.567] and [-0.756, 0.331, -0.567].