If vector v has an initial point at (2, 3) and a terminal point at (6, 6) and vector w has an initial point at (3, 2) and a terminal point at (6, 6), are the two vectors equivalent?
A) Yes, they have the same magnitude and same direction.
B) No, they have the same direction but different magnitude.
C) No, they have the same magnitude but different direction.
D) No, they have different magnitude and different direction.
E) Yes, they have the same magnitude and the direction does not matter.
To determine if the two vectors v and w are equivalent, we need to compare their magnitudes and directions.
The magnitude of a vector can be calculated using the distance formula:
Magnitude of v = sqrt((6-2)^2 + (6-3)^2) = sqrt(16 + 9) = sqrt(25) = 5
Magnitude of w = sqrt((6-3)^2 + (6-2)^2) = sqrt(9 + 16) = sqrt(25) = 5
The magnitudes of both vectors are the same.
To determine if they have the same direction, we can compare the slopes of the lines passing through their initial and terminal points.
The slope of the line passing through the points (2, 3) and (6, 6) is given by (6-3)/(6-2) = 3/4.
The slope of the line passing through the points (3, 2) and (6, 6) is given by (6-2)/(6-3) = 4/3.
Since the slopes are not equal, the two vectors v and w have different directions.
Therefore, the correct answer is:
D) No, they have different magnitude and different direction.
To determine if two vectors are equivalent, we need to compare their magnitudes and directions.
For vector v, the initial point is (2, 3) and the terminal point is (6, 6). To find the magnitude of vector v, we can use the distance formula:
Magnitude of vector v = √((6 - 2)^2 + (6 - 3)^2) = √(4^2 + 3^2) = √(16 + 9) = √25 = 5
The direction of vector v can be determined by subtracting the coordinates of the initial point from the coordinates of the terminal point:
Direction of vector v = (6 - 2, 6 - 3) = (4, 3)
For vector w, the initial point is (3, 2) and the terminal point is (6, 6). To find the magnitude of vector w, we can use the distance formula:
Magnitude of vector w = √((6 - 3)^2 + (6 - 2)^2) = √(3^2 + 4^2) = √(9 + 16) = √25 = 5
The direction of vector w can be determined by subtracting the coordinates of the initial point from the coordinates of the terminal point:
Direction of vector w = (6 - 3, 6 - 2) = (3, 4)
Comparing the magnitudes and directions of vector v and vector w, we can see that they both have the same magnitude (5), but their directions are different.
Therefore, the answer is:
D) No, they have different magnitude and different direction.
vector v = <(6-2,6-3)> = <4,3)
vector w = <3,4>
magnitude v = √(16+9) = 5
magnitude w = √(9+16) = 5
what about their slopes?