Two vectors have magnitudes 3.0 and 4.0. How are the directions of the two vectors related given the following conditions?
1) The sum has magnitude 7.0.
(a)They both point in the same direction.
(b)They each point in opposite directions.
(c) They are perpendicular to each other.
2) If the sum has magnitude 5.0.
(a)They both point in the same direction.
(b)They each point in opposite direction.
(c) They are perpendicular to each other.
3) What relationship between the directions gives the smallest magnitude sum?
(a)They both point in the same direction.
(b)They each point in opposite direction.
(c) They are perpendicular to each other.
What is this magnitude?
If you haven't had trig, you can do this using a paper and pencil and a ruler.
1. Draw a horizontal arrow 3.0 inches long, with the arrow head on the right end.
2. Starting at the above arrow's arrowhead, draw another arrow, this time 4.0 inches long, in all 3 possible directions.
3. You are now interested in the total displacement: that is, the length of a line if drawn from the very beginning of the first arrow (the end opposite its arrowhead) to the very end of the second arrow (the tip of its arrowhead).
Now just measure each displacement and answer the questions.
Forgot to say:
When drawing vectors as arrows, the magnitude is the length of the arrow.
(The second component of a vector is direction.)
1) If the sum of the two vectors has magnitude 7.0, then the vectors are pointing in the same direction. Choice (a) is correct.
2) If the sum of the two vectors has magnitude 5.0, then the vectors are pointing in opposite directions. Choice (b) is correct.
3) To achieve the smallest magnitude sum, the two vectors should be perpendicular to each other. Choice (c) is correct.
The magnitude of the sum is given as 7.0 in question 1 and as 5.0 in question 2. However, the magnitude is not mentioned for question 3.
To find the relationship between the directions of two vectors given certain conditions, we can use vector addition. The magnitude of the sum of two vectors can be found using the formula:
|A + B| = sqrt((|A|^2) + (|B|^2) + (2 * |A| * |B| * cosθ))
where A and B are the magnitudes of the two vectors, and θ is the angle between them.
Let's analyze each condition and determine the relationship between the directions of the vectors:
1) The sum has magnitude 7.0.
Using the formula mentioned above, we have:
7.0 = sqrt((3.0^2) + (4.0^2) + (2 * 3.0 * 4.0 * cosθ))
Simplifying this equation:
49.0 = 25 + 16 + 24 * cosθ
49.0 = 41 + 24 * cosθ
8.0 = 24 * cosθ
cosθ = 8.0 / 24
cosθ = 1/3
Since the cosine of θ is positive, θ must be an acute angle. Therefore, the vectors must point in:
(a) They both point in the same direction.
2) If the sum has magnitude 5.0.
Using the same formula:
5.0 = sqrt((3.0^2) + (4.0^2) + (2 * 3.0 * 4.0 * cosθ))
Simplifying this equation:
25.0 = 25 + 16 + 24 * cosθ
25.0 = 41 + 24 * cosθ
-16 = 24 * cosθ
cosθ = -16 / 24
cosθ = -2/3
Here, since the cosine of θ is negative, θ must be an obtuse angle. Therefore, the vectors must point in:
(b) They each point in opposite directions.
3) What relationship between the directions gives the smallest magnitude sum?
To find the smallest magnitude sum, we need to minimize the value of cosθ. The smallest value for cosine occurs when θ is 90 degrees or perpendicular to each other. Therefore, the vectors must point in:
(c) They are perpendicular to each other.
Finally, the magnitude of the sum is given by the equations we derived earlier:
1) When the sum has magnitude 7.0.
|A + B| = sqrt((3.0^2) + (4.0^2) + (2 * 3.0 * 4.0 * (1/3)))
|A + B| = sqrt(9 + 16 + (24/3))
|A + B| = sqrt(9 + 16 + 8)
|A + B| = sqrt(33)
2) When the sum has magnitude 5.0.
|A + B| = sqrt((3.0^2) + (4.0^2) + (2 * 3.0 * 4.0 * (-2/3)))
|A + B| = sqrt(9 + 16 + (-16/3))
|A + B| = sqrt(25 - (16/3))
|A + B| = sqrt((75/3) - (16/3))
|A + B| = sqrt(59/3)
Therefore, the magnitudes are:
1) |A + B| = sqrt(33) ≈ 5.74
2) |A + B| = sqrt(59/3) ≈ 4.26