Subtracting Vectors: How to Find Magnitude of A-B

[nrdiamon]

This is the problem that I was given:

Given two vectors A and B, with magnitudes |A| = 45.7 and |B|=38.2 and directions (from the x-axis) θA=64° and θB=145°, find the magnitude of (A-B)

I know that this somehow involves triangles and trigonometry, but I am really confused as to how I do this. Please help!

 

[berkeman]

This is the problem that I was given:

Given two vectors A and B, with magnitudes |A| = 45.7 and |B|=38.2 and directions (from the x-axis) θA=64° and θB=145°, find the magnitude of (A-B)

I know that this somehow involves triangles and trigonometry, but I am really confused as to how I do this. Please help!

Welcome to the PF. To add/subtract vectors, you will use rectangular coordinates and rectangular components, and then convert back into polar form if needed for the final answer.

Does that make sense? If not, go to wikipedia.org, and search on vector rectangular polar conversion. Actually a better match at wiki is vector polar coordinate system...

 

[tomcue]

Subtracting vectors involves the process of vector addition with the second vector being reversed in direction. In this case, we are looking to find the magnitude of (A-B), which can be represented as |A-B|. To do this, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, we can use the magnitudes of A and B as the sides of the triangle and the magnitude of (A-B) as the hypotenuse.

To find the magnitude of (A-B), we can first find the components of A and B using the given directions. The x component of A can be found by multiplying the magnitude of A by the cosine of θA, and the y component can be found by multiplying the magnitude of A by the sine of θA. Similarly, the x and y components of B can be found using the magnitudes and directions given for B.

Once we have the x and y components of A and B, we can subtract them to find the components of (A-B). The x component of (A-B) would be A_x - B_x, and the y component would be A_y - B_y. We can then use the Pythagorean theorem to find the magnitude of (A-B) by taking the square root of the sum of the squares of the x and y components.

In this specific problem, the x component of A would be 45.7 * cos(64°) = 22.87 and the y component would be 45.7 * sin(64°) = 40.67. Similarly, the x component of B would be 38.2 * cos(145°) = -16.93 and the y component would be 38.2 * sin(145°) = 32.54. Therefore, the x component of (A-B) would be 22.87 - (-16.93) = 39.8 and the y component would be 40.67 - 32.54 = 8.13.

Using the Pythagorean theorem, the magnitude of (A-B) would be √(39.8^2 + 8.13^2) = 40.6. Therefore, the magnitude of (A-B) is approximately