For the following vector addition diagrams, use Pythagorean Theorem to determine the magnitude of the resultant. Use SOH CAH TOA to determine the direction. Use the tail end relative to due east.
^
I 4 m
I
------>I
3 m
I've got the Pythagoren Theorem and the angle measure down. I just don't understand the addition and tail end relative to east thing.
the 4m line should intersect with the arrow on the right of the 3m line.
Nope. head to tail is the rule. Move the 3m arrow up to start at the head(top) of the 4m arrow. Then the resultant is from the origin to the head of the 3m arrow.
3m
----->o
^
|
|4m The resulant goes from x to o
|
|
x
To determine the magnitude of the resultant vector using the Pythagorean Theorem, you will need to find the sum of the squares of the individual vector magnitudes, and then take the square root of the sum. In this case, you have two vectors:
1. A vector with a magnitude of 3 m, pointing to the right.
2. A vector with a magnitude of 4 m, pointing upward.
Using Pythagorean Theorem, you can find the magnitude of the resultant vector using the equation:
Resultant magnitude = √(3^2 + 4^2)
Simplifying this, we get:
Resultant magnitude = √(9 + 16) = √25 = 5 m
Now, let's determine the direction of the resultant vector using the "tail end relative to due east" approach.
- Start with the vector pointing to the right (3 m).
- Then draw the vector pointing upward (4 m).
- Connect the tail of the first vector to the head of the second vector.
The resultant vector will be the vector that starts from the tail of the first vector and ends at the head of the second vector. In this case, it will be a diagonal line connecting the tail of the first vector to the head of the second vector.
To determine the direction of this resultant vector, we use the "tail end relative to due east" method. This means we consider the angle between the resultant vector and the positive x-axis (east direction).
Since the resultant vector is pointing up and to the right, the angle between the resultant vector and the positive x-axis (east) will be an acute angle. We can use SOH CAH TOA to determine this angle. Here's how:
- In a right-angled triangle, "S" stands for sine, "C" stands for cosine, and "T" stands for tangent.
- In this case, we have the opposite side (4 m) and the adjacent side (3 m).
- We want to find the angle between the resultant vector and the positive x-axis, so we need to use the ratio of the opposite and adjacent sides, which is tangent (T).
Using the equation:
T = opposite/adjacent,
T = 4/3.
Now you can use the inverse tangent function (tan^(-1)) to find the angle:
Angle = tan^(-1)(4/3)
Using a calculator to find the inverse tangent of (4/3), you will get the angle:
Angle ≈ 53.13 degrees
So, the magnitude of the resultant vector is 5 m and the direction is approximately 53.13 degrees relative to due east.