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)
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.
To determine whether a vector is parallel or perpendicular to another vector, we can use the dot product. If the dot product is zero, the vectors are perpendicular. If the dot product is nonzero, the vectors are parallel.
Let's calculate the dot product with each vector:
A) (2, 2, -2) dot (1, 1, -1) = (2*1) + (2*1) + (-2*-1) = 2 + 2 + 2 = 6
B) (2, -2, 0) dot (1, 1, -1) = (2*1) + (-2*1) + (0*-1) = 2 - 2 + 0 = 0
C) (-2, 2, 2) dot (1, 1, -1) = (-2*1) + (2*1) + (2*-1) = -2 + 2 - 2 = -2
D) (1, 0, 1) dot (1, 1, -1) = (1*1) + (0*1) + (1*-1) = 1 + 0 - 1 = 0
Based on the dot products:
A) The dot product is nonzero (6), so (2, 2, -2) is parallel to (1, 1, -1).
B) The dot product is zero (0), so (2, -2, 0) is perpendicular to (1, 1, -1).
C) The dot product is nonzero (-2), so (-2, 2, 2) is parallel to (1, 1, -1).
D) The dot product is zero (0), so (1, 0, 1) is perpendicular to (1, 1, -1).
Therefore, vectors A, C are parallel to (1, 1, -1), and vectors B, D are perpendicular to (1, 1, -1).