To determine whether vectors are parallel or perpendicular, we can take the dot product of the two vectors.
The dot product of two vectors u = (u1, u2, u3) and v = (v1, v2, v3) is given by u · v = u1v1 + u2v2 + u3v3.
If u · v = 0, the vectors are perpendicular.
If u · v ≠ 0, the vectors are not perpendicular.
Now let's test each vector:
A) (2, 2, -2)
(1, 1, -1) · (2, 2, -2) = 1(2) + 1(2) + (-1)(-2) = 2 + 2 + 2 = 6
Since the dot product is not 0, vector A is not perpendicular to (1, 1, -1).
B) (2, -2, 0)
(1, 1, -1) · (2, -2, 0) = 1(2) + 1(-2) + (-1)(0) = 2 - 2 + 0 = 0
The dot product is 0, so vector B is perpendicular to (1, 1, -1).
C) (-2, 2, 2)
(1, 1, -1) · (-2, 2, 2) = 1(-2) + 1(2) + (-1)(2) = -2 + 2 - 2 = -2
The dot product is not 0, so vector C is not perpendicular to (1, 1, -1).
D) (1, 0, 1)
(1, 1, -1) · (1, 0, 1) = 1(1) + 1(0) + (-1)(1) = 1 + 0 - 1 = 0
The dot product is 0, so vector D is perpendicular to (1, 1, -1).
E) (missing vector)
Since the vector is missing, we cannot determine if it is parallel or perpendicular to (1, 1, -1).
F) (missing vector)
Since the vector is missing, we cannot determine if it is parallel or perpendicular to (1, 1, -1).
In conclusion:
Vector B, (2, -2, 0), is perpendicular to (1, 1, -1).
Vectors A, C, D, E, and F, cannot be determined if they are parallel or perpendicular to (1, 1, -1) based on the information given.
Which of he following vectors are parallel or perpendicular to (1, 1, -1)?
A)(2, 2, -2) d) (1, 0, 1)
B)(2, -2, 0) e)
C)(-2, 2, 2) f)
1 answer