How to Find a Vector’s Magnitude and Direction – dummies

theta = tan–1(y/x)

\r\nSuppose that the coordinates of the vector are ( 3, 4 ). You can find the slant theta as the tan –1 ( 4/3 ) = 53 degrees.\r\n\r\nYou can use the Pythagorean theorem to find the hypotenuse — the magnitude, v — of the triangle formed by x, y, and v: \r\n\r\n \r\n\r\nPlug in the numbers for this example to get\r\n\r\n \r\n\r\nSo if you have a vector given by the coordinates ( 3, 4 ), its magnitude is 5, and its fish is 53 degrees.\r\n

Sample question

\r\n

\r\n \t

1. \r\n

Convert the vector given by the coordinates (1.0, 5.0) into magnitude/angle format.

\r\n

The correct answer is magnitude 5.1, angle 79 degrees.

\r\n\r\n

\r\n \t

1. \r\n

Apply the Pythagorean theorem to find the magnitude. Plug in the numbers to get 5.1.

\r\n

2. \r\n \t

3. \r\n

Apply the equation theta= tan–1(y/x) to find the angle. Plug in the numbers to get tan–1(5.0/1.0) = 79 degrees.

\r\n

4. \r\n

\r\n

2. \r\n

\r\n

Practice questions

\r\n

\r\n \t

1. \r\n

Convert the vector (5.0, 7.0) into magnitude/angle form.

\r\n

2. \r\n \t

3. \r\n

Convert the vector (13.0, 13.0) into magnitude/angle form.

\r\n

4. \r\n \t

5. \r\n

Convert the vector (–1.0, 1.0) into magnitude/angle form.

\r\n

6. \r\n \t

7. \r\n

Convert the vector (–5.0, –7.0) into magnitude/angle form.

\r\n

8. \r\n

\r\nFollowing are answers to the practice questions : \r\n

\r\n \t

1. \r\n

Magnitude 8.6, angle 54 degrees

\r\n\r\n

\r\n \t

1. \r\n

Apply the equation

\r\n

2. \r\n

\r\n

to find the magnitude, which is 8.6.

\r\n\r\n

\r\n \t

1. \r\n

Apply the equation theta = tan–1(y/x) to find the angle: tan–1(7.0/5.0) = 54 degrees.

\r\n

2. \r\n

\r\n

2. \r\n \t

3. \r\n

Magnitude 18.4, angle 45 degrees

\r\n\r\n

\r\n \t

1. \r\n

Apply the equation

\r\n

2. \r\n

\r\n

to find the magnitude, which is 18.4.

\r\n\r\n

\r\n \t

1. \r\n

Apply the equation theta = tan–1(y/x) to find the angle: tan–1(13.0/13.0) = 45 degrees.

\r\n

2. \r\n

\r\n

4. \r\n \t

5. \r\n

Magnitude 1.4, angle 135 degrees

\r\n\r\n

\r\n \t

1. \r\n

Apply the equation

\r\n\r\n

to find the magnitude, which is 1.4.

\r\n

2. \r\n \t

3. \r\n

Apply the equation theta = tan–1(y/x) to find the angle: tan–1(1.0/–1.0) = –45 degrees.

\r\n

However, note that the angle must really be between 90 degrees and 180 degrees because the first vector component is negative and the second is positive. That means you should add 180 degrees to –45 degrees, giving you 135 degrees (the tangent of 135 degrees is also 1.0/–1.0 = –1.0).

\r\n

4. \r\n

\r\n

6. \r\n \t

7. \r\n

Magnitude 8.6, angle 234 degrees

\r\n\r\n

\r\n \t

1. \r\n

Apply the equation

\r\n\r\n

to find the magnitude, which is 8.6.

\r\n

2. \r\n \t

3. \r\n

Apply the equation theta = tan–1(y/x) to find the angle: tan–1(–7.0/–5.0) = 54 degrees.

\r\n

However, note that the angle must really be between 180 degrees and 270 degrees because both vector components are negative. That means you should add 180 degrees to 54 degrees, giving you 234 degrees (the tangent of 234 degrees is also –7.0/–5.0 = 7.0/5.0).

\r\n

4. \r\n

\r\n

8. \r\n

“, ” description ” : ” In physics, when you ’ re given the vector components, such as ( 3, 4 ), you can easily convert to the magnitude/angle means of expressing vectors using trigonometry.\r\n\r\nFor example, take a count at the vector in the image.\r\n\r\n\r\n\r\nSuppose that you ’ re given the coordinates of the end of the vector and want to find its magnitude, v, and fish, theta. Because of your cognition of trigonometry, you know\r\n\r\n\r\n\r\nWhere tan theta is the tangent of the angle. This means that\r\n

theta = tan–1(y/x)

\r\nSuppose that the coordinates of the vector are ( 3, 4 ). You can find the slant theta as the tangent –1 ( 4/3 ) = 53 degrees.\r\n\r\nYou can use the Pythagorean theorem to find the hypotenuse — the magnitude, v — of the triangle formed by x, y, and v: \r\n\r\n\r\n\r\nPlug in the numbers for this example to get\r\n\r\n\r\n\r\nSo if you have a vector given by the coordinates ( 3, 4 ), its magnitude is 5, and its fish is 53 degrees.\r\n

Sample question

\r\n

\r\n \t

1. \r\n

Convert the vector given by the coordinates (1.0, 5.0) into magnitude/angle format.

\r\n

The correct answer is magnitude 5.1, angle 79 degrees.

\r\n\r\n

\r\n \t

1. \r\n

Apply the Pythagorean theorem to find the magnitude. Plug in the numbers to get 5.1.

\r\n

2. \r\n \t

3. \r\n

Apply the equation theta= tan–1(y/x) to find the angle. Plug in the numbers to get tan–1(5.0/1.0) = 79 degrees.

\r\n

4. \r\n

\r\n

2. \r\n

\r\n

Practice questions

\r\n

\r\n \t

1. \r\n

Convert the vector (5.0, 7.0) into magnitude/angle form.

\r\n

2. \r\n \t

3. \r\n

Convert the vector (13.0, 13.0) into magnitude/angle form.

\r\n

4. \r\n \t

5. \r\n

Convert the vector (–1.0, 1.0) into magnitude/angle form.

\r\n

6. \r\n \t

7. \r\n

Convert the vector (–5.0, –7.0) into magnitude/angle form.

\r\n

8. \r\n

\r\nFollowing are answers to the practice questions : \r\n

\r\n \t

1. \r\n

Magnitude 8.6, angle 54 degrees

\r\n\r\n

\r\n \t

1. \r\n

Apply the equation

\r\n

2. \r\n

\r\n

to find the magnitude, which is 8.6.

\r\n\r\n

\r\n \t

1. \r\n

Apply the equation theta = tan–1(y/x) to find the angle: tan–1(7.0/5.0) = 54 degrees.

\r\n

2. \r\n

\r\n

2. \r\n \t

3. \r\n

Magnitude 18.4, angle 45 degrees

\r\n\r\n

\r\n \t

1. \r\n

Apply the equation

\r\n

2. \r\n

\r\n

to find the magnitude, which is 18.4.

\r\n\r\n

\r\n \t

1. \r\n

Apply the equation theta = tan–1(y/x) to find the angle: tan–1(13.0/13.0) = 45 degrees.

\r\n

2. \r\n

\r\n

4. \r\n \t

5. \r\n

Magnitude 1.4, angle 135 degrees

\r\n\r\n

\r\n \t

1. \r\n

Apply the equation

\r\n\r\n

to find the magnitude, which is 1.4.

\r\n

2. \r\n \t

3. \r\n

Apply the equation theta = tan–1(y/x) to find the angle: tan–1(1.0/–1.0) = –45 degrees.

\r\n

However, note that the angle must really be between 90 degrees and 180 degrees because the first vector component is negative and the second is positive. That means you should add 180 degrees to –45 degrees, giving you 135 degrees (the tangent of 135 degrees is also 1.0/–1.0 = –1.0).

\r\n

4. \r\n

\r\n

6. \r\n \t

7. \r\n

Magnitude 8.6, angle 234 degrees

\r\n\r\n

\r\n \t

1. \r\n

Apply the equation

\r\n\r\n

to find the magnitude, which is 8.6.

\r\n

2. \r\n \t

3. \r\n

Apply the equation theta = tan–1(y/x) to find the angle: tan–1(–7.0/–5.0) = 54 degrees.

\r\n

However, note that the angle must really be between 180 degrees and 270 degrees because both vector components are negative. That means you should add 180 degrees to 54 degrees, giving you 234 degrees (the tangent of 234 degrees is also –7.0/–5.0 = 7.0/5.0).

\r\n

4. \r\n

\r\n

8. \r\n

“, ” endorsement ” : ” ”, ” authors ” : [ { “ authorId ” :8967, ” diagnose ” : ” Steven Holzner ”, ” type slug ” : ” steven-holzner ”, ” description ” : ”

Dr. Steven Holzner has written more than 40 books about physics and programming. He was a contributing editor at PC Magazine and was on the faculty at both MIT and Cornell. He has authored Dummies titles including Physics For Dummies and Physics Essentials For Dummies. Dr. Holzner received his PhD at Cornell.

Joseph A. Allen, PhD is a professor of industrial and organizational (I/O) psychology at the University of Utah. His articles have appeared in Human Relations, Journal of Business Psychology, and more.