Vector {A} points in the negative y direction and has a magnitude of 14 units. Vector {B} has twice the magnitude and points in the positive x direction.
Find the direction and magnitude of {A}+{B} counterclockwise from the positive x axis.
If you add them, magnitude is 14sqrt5
direction: ArcTan 14/28=arctan .5=90+26.56 degrees
To find the direction and magnitude of the result of adding vector {A} and vector {B}, we can use vector addition.
First, let's analyze the given information about vector {A} and vector {B}:
1. Vector {A} points in the negative y direction, which means it has an angle of 270 degrees counterclockwise from the positive x-axis.
2. Vector {A} has a magnitude of 14 units.
3. Vector {B} has twice the magnitude of vector {A}, which means it has a magnitude of 28 units.
4. Vector {B} points in the positive x direction, which means it has an angle of 0 degrees counterclockwise from the positive x-axis.
Now, let's find the result of adding vector {A} and vector {B}:
1. To add vector {A} and vector {B}, we add their x-components and y-components separately.
- The x-component of vector {A} is 0 because it points strictly in the y-direction.
- The y-component of vector {A} is -14 because it has a magnitude of 14 units in the negative y-direction.
- The x-component of vector {B} is 28 because it points strictly in the x-direction.
- The y-component of vector {B} is 0 because it points strictly in the x-direction.
2. Now, add the x-components and y-components to find the result:
- The x-component of the result is 0 + 28 = 28.
- The y-component of the result is -14 + 0 = -14.
3. To find the magnitude of the result, use the Pythagorean theorem:
- Magnitude = √(x^2 + y^2) = √(28^2 + (-14)^2) = √(784 + 196) = √980 = 14√5 units.
4. To find the direction counterclockwise from the positive x-axis, we need to calculate the angle using trigonometry. Since the x-component is positive and the y-component is negative, the angle is in the fourth quadrant.
- Angle = arctan(y-component / x-component) = arctan(-14 / 28) = arctan(-0.5).
- Using a calculator or reference table, the angle is approximately -26.565 degrees.
Therefore, the direction and magnitude of {A} + {B} counterclockwise from the positive x-axis is approximately 14√5 units at an angle of -26.565 degrees counterclockwise.