What are the Magnitude and Angle of Vectors in Terms of their Components?

• Susanem7389
In summary, the formula for finding the magnitude of a vector is A = √(Ax² + Ay²), and the formula for finding the angle is angle = tan⁻¹(Ay/Ax). However, if the x-component is negative, 180 degrees must be added to the angle calculated by the formula. It is important to also understand the concept of vector angles being measured in a counter-clockwise direction from the +x direction on a coordinate plane. Drawing out the vectors on a coordinate plane can help visualize and solve for the magnitude and angle.
Susanem7389
Rewrite the following vectors in terms of their magnitude and angle (counterclockwise from the +x direction)
a) A velocity vector with an x component of -75 m/s and a y component of 35 m/s
- I found the magnitude by using A= square root of ( Ax squared plus Ay squared ), however I did not get the correct answer for the angle. I used the formula angle = tan -1 ( Ay/ Ax)

b) A force vector with a magnitude of 50 lb that is in the third quadrant with an x component whose magnitude is 40 lb.
- I could not find the correct magnitude and angle with the same formula used in part A.

Could you show your angel for part a) as well as magnitude and angel for part b)?

Note that your calculator cannot distinguish arctan(1/1) from arctan(-1/-1).
Rule of thumb... if the x-component is negative, add 180 degrees to what your calculator tells you when using arctan. (Some calculators may have something like an atan2 function.)

For part A, adding 180 degrees to what the calculator gave me, I got the correct answer, however it did not work for part b.

Susanem7389 said:
For part A, adding 180 degrees to what the calculator gave me, I got the correct answer, however it did not work for part b.

You'll have to show your work...

I would strongly suggest drawing out each of these vectors on an x-y coordinate plane so you can see exactly what the angles the formulas are giving you. The vector angle is always going to be measured with respect to the +x direction, in a counter-clockwise fashion; in other words, a vector in this +x direction would have an angle of 0 degrees.

Try to not just remember formulas, look at the right triangles the vector magnitude, x-component, and y-component are forming. The magnitude is going to be the hypotenuse of the right triangle that is formed.

for b) They give the the magnitude of the vector already, you just need to find the angle. Remember that the angle will be measure from the +x direction rotating counter-clockwise until it meets the vector in question.

1. What is a vector?

A vector is a quantity that has both magnitude and direction. It is represented by an arrow, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction.

2. What is the magnitude of a vector?

The magnitude of a vector is the length of the vector. It is a scalar quantity and is always positive. It can be calculated using the Pythagorean theorem, by taking the square root of the sum of the squares of the vector's components.

3. How is the angle of a vector measured?

The angle of a vector is measured counterclockwise from the positive x-axis. This is also known as the standard position of the vector. It is typically measured in degrees or radians.

4. How do you find the magnitude of a vector using its components?

If the vector's components are given in terms of its x and y axes, the magnitude can be calculated using the Pythagorean theorem. The formula is magnitude = √(x² + y²).

5. How do you find the angle of a vector using its components?

The angle of a vector can be calculated using trigonometric functions. The formula is angle = arctan(y/x), where y is the vertical component and x is the horizontal component. This will give the angle in radians, which can be converted to degrees if needed.

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