You've already learned how to add and subtract vectors algebraically by adding subtacting their components. You've also learned how to calculate the magnitude of a vector from its components. But what if you don't know the components of the vector? For example, what if you are given a displacement vector of magnitude 10.0 m at an angle of 30° from the horizontal? How can you get the components from this information? Alternatively, how can you use vector components and its magnitude to calculate the angle it makes with a horizontal axis?
Remeber that a vector and its components form a right triangle, like the one shown below.
If the angle between the vector and the x-axis is θ, then using the definitions of the trig. functions sine, cosine, and tangent, we have write the relationships between the components and the magnitude of the vector (which is the hypotenuse of the right triangle).
It's vital that you know the unit circle shown below.
A vector pointing in the +x direction makes an angle of 0° with the +x axis.
A vector pointing in the +y direction makes an angle of 90° with the +x axis.
A vector pointing in the -x direction makes an angle of 180° with the +x axis.
A vector pointing in the -y direction makes an angle of 270° with the +x axis.
A vector pointing at any angle to the right of the origin will have a positive x-component.
A vector pointing any angle to the left of the origin will have a negative x-component.
A vector pointing at any angle upward from the origin will have a positive y-component.
A vector pointing at any angle downward from the origin will have a negative y-component.
The only way to be comfortable with vectors and trigonometery is to practice. At first it is confusing; only practice remedies the confusion.
Note that θ is the angle that the vector makes with the +x axis (also called the horizontal axis although technically it can be defined in any orientation you wish). The equations above that relate the vector components to the vector magnitude and θ are only correct for this definition of θ. If an angle is defined any other way (perhaps with respect to the +y axis), then you have to either figure out what θ is, or you have to use the trig. functions to find new relationships between the vector components and the vector magnitude (hypotenuse).
Angles can be reported going counterclockwise or clockwise around the unit circult. If you report your angle as the angle below, or clockwise from the +x axis, then you must report it as a negative angle. For example, 270° is the same as -90°.
For all of the questions below, please sketch a picture of the vector and the coordinate system. It helps!
1. (a) A displacement vector has a magnitude of 100 m and an angle of 110° with respect to the +x axis. What are its x and y components?
&Delta x = <_> m &Delta y = <_> m
(b) What angle does that vector make with the +y axis? <_> °
2. (a) At some instant, a car has a velocity of 20.0 m/s in a direction of 15° West of South. (In other words, if N is the +y axis and E is the +x axis, this angle is given with respect to the -y axis.) What is the angle of the vector with respect to the +x axis? <_> °
(b) What are the x and y components of the velocity of the car?
vx = <_> m/s vy = <_> m/s
3. The acceleration at some instant of a runner as she rounds the curve on a track is <-1.5 , 0.5> m/s2.What is the magnitude of the acceleration of the runner and what angle does this vector make with respect to the +x axis?
θ = <_> °