Final answer:
a. The quaternion that rotates the vector v by 90 degrees around the z-axis is (0, 0, 0, 1). b. The quaternion that rotates the vector v by 90 degrees around the y-axis is (0, 0.707, 0, 0.707). c. The combination of the quaternion from part a and part b results in the vector (0, 2, 2, 0). d. The quaternion that achieves the same rotation results as the combination of the quaternions from part a and part b is (0, 0.707, 0, 0.707).
Step-by-step explanation:
a. Quaternion Rotation Around the z-axis
To rotate the vector v = (2, 2, 0) by 90 degrees around the z-axis, we can use the quaternion q = (0, 0, 0, 1). This quaternion represents a rotation of 90 degrees around the z-axis. To rotate a vector using a quaternion, we first convert the vector into a quaternion by setting the real component to 0 and using the vector components as the imaginary components. Then, we multiply the quaternion representing the vector by the quaternion representing the rotation. The resulting quaternion can be converted back into a vector by taking its imaginary components.
Using the formula v' = qvq', where v' is the rotated vector, q is the rotation quaternion, and q' is the conjugate of q, we can calculate the rotated vector:
v' = qvq':
v' = (0, 0, 0, 1)(0, 2, 2, 0)(0, 0, 0, -1)
v' = (0, 0, -2, 2)
b. Quaternion Rotation Around the y-axis
To rotate the vector v = (2, 2, 0) by 90 degrees around the y-axis, we can use the quaternion q = (0, 0.707, 0, 0.707). This quaternion represents a rotation of 90 degrees around the y-axis. Following the same steps as in part a, we can calculate the rotated vector:
v' = qvq':
v' = (0, 0.707, 0, 0.707)(0, 2, 2, 0)(0, -0.707, 0, 0.707)
v' = (2, 2, 0)
c. Rotation Using Quaternion from Part a and Part b
To rotate the vector v using the quaternion from part a and then the quaternion from part b, we can simply apply the formula v'' = q''v'q'', where v'' is the final rotated vector, q'' is the quaternion from part b, and v' is the vector rotated using the quaternion from part a:
v'' = q''v'q'':
v'' = (0, 0.707, 0, 0.707)(2, 2, 0)(0, -0.707, 0, 0.707)
v'' = (0, 2, 2, 0)
d. Combination of Quaternions from Part a and Part b
To find the quaternion that achieves the same rotation results as the combination of the quaternions from part a and part b, we can simply multiply the two quaternions together:
q_combined = q_b * q_a:
q_combined = (0, 0.707, 0, 0.707)(0, 0, 0, 1)
q_combined = (0, 0.707, 0, 0.707)
Therefore, the quaternion that can be used to rotate the vector v to the result obtained in part c is q_combined = (0, 0.707, 0, 0.707).