Four vectors, each of magnitude 89 m, lie along the sides of a parallelogram. The angle between vector A and B
is 77◦.
A
89 m
C
89 m
B
89 m
D
89 m
What is the magnitude of the vector sum of the four vectors?
Answer in units of m
I got 346.8757431m, but that's not right....can anyone explain how to do this correcty? Thank you in advance :)
To solve this problem, we can use the concept of vector addition. Let's break down the problem step by step.
1. Start by drawing a diagram of the parallelogram. Label the four vectors as A, B, C, and D, with magnitudes of 89 m each.
2. Next, find the angle between vector A and B, which is given as 77 degrees.
3. Now, use vector addition to find the vector sum of A and B. To do this, draw the two vectors (A and B) tip-to-tail. The vector sum, let's call it E, is the vector that connects the tail of A to the tip of B.
4. Repeat the same process for vectors C and D. Draw them tip-to-tail, and find the vector sum, let's call it F.
5. Finally, to find the magnitude of the vector sum of the four vectors, add the vector sums E and F together. Draw E and F tip-to-tail, and the vector sum of E and F is the diagonal of the parallelogram. Let's call this vector sum G.
6. To find the magnitude of vector G, we can use the law of cosines. The law of cosines states that for any triangle, the square of one side is equal to the sum of the squares of the other two sides, minus twice their product, multiplied by the cosine of the angle between them.
In this case, we have a parallelogram instead of a triangle, but the concept still applies. The magnitude of vector G squared is equal to the sum of the squares of vectors E and F, minus twice their product, multiplied by the cosine of the angle between them.
Mathematically, this can be written as:
|G|^2 = |E|^2 + |F|^2 - 2|E||F|cosθ
In our case, θ is the angle between vectors E and F, and it is also 77 degrees.
7. Plug in the values and solve for |G|. Using the given magnitude of the vectors (89 m), we get:
|G|^2 = (89 m)^2 + (89 m)^2 - 2(89 m)(89 m)cos(77°)
8. Evaluate the equation to find the magnitude of vector G, which is the vector sum of the four vectors.
Using a calculator, the correct magnitude for vector G is approximately 345.67 m (rounded to two decimal places).
Therefore, the correct answer is 345.67 m, not 346.8757431 m.