Better recognize your Trig.
You are watching: What angle does this vector make with the x axis?
You"ve already learned just how to add and subtract vectors algebraically by including subtacting your components. You"ve also learned how to calculate the magnitude of a vector indigenous its components. But what if girlfriend don"t understand the components of the vector? because that example, what if you are offered a displacement vector of size 10.0 m at an angle of 30° from the horizontal? How have the right to you get the materials from this information? Alternatively, how deserve to you use vector components and also its magnitude to calculate the edge it provides with a horizontal axis?
Remeber that a vector and also its components type a right triangle, favor the one shown below.
If the angle between the vector and the x-axis is θ, then using the definitions of the trig. Attributes sine, cosine, and also tangent, we have actually write the relationships between the components and also the size of the vector (which is the hypotenuse of the ideal triangle).
It"s vital that you understand the unit circle presented below.
A vector pointing in the +x direction renders an angle of 0° through the +x axis.A vector pointing in the +y direction provides an edge of 90° v the +x axis.A vector pointing in the -x direction makes an angle of 180° through the +x axis.A vector pointing in the -y direction provides an angle of 270° through the +x axis.A vector pointing at any type of angle come the right of the origin will have actually a hopeful x-component.A vector pointing any kind of angle come the left of the origin will have actually a an unfavorable x-component.A vector pointing at any kind of angle increase from the origin will have a hopeful y-component.A vector pointing at any angle bottom from the beginning will have a negative y-component.
The only way to be comfortable with vectors and also trigonometery is to practice. At an initial it is confusing; just practice remedies the confusion.
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Note the θ is the angle that the vector renders with the +x axis (also dubbed the horizontal axis although technically it deserve to be characterized in any type of orientation friend wish). The equations above that called the vector contents to the vector magnitude and also θ are only correct for this definition of θ. If an edge is defined any type of other way (perhaps with respect come the +y axis), climate you have to either figure out what θ is, or you have to use the trig. Attributes to find new relationships between the vector components and also the vector size (hypotenuse).
Angles deserve to be reported going counterclockwise or clockwise about the unit circult. If you report her angle as the edge below, or clockwise from the +x axis, then you have to report it together a an unfavorable angle. For example, 270° is the very same as -90°.CheckPointFor every one of the inquiries below, please sketch a photo of the vector and also the coordinate system. That helps!
1. (a) A displacement vector has a magnitude of 100 m and an edge of 110° through respect to the +x axis. What space its x and y components?
&Delta x = m &Delta y = m
(b) What edge does that vector make with the +y axis? °
2. (a) At part instant, a auto has a velocity of 20.0 m/s in a direction that 15° West the South. (In various other words, if N is the +y axis and E is the +x axis, this edge is given with respect come the -y axis.) What is the angle of the vector with respect to the +x axis? °
(b) What space the x and also y contents of the velocity that the car?
vx = m/s vy = m/s
3. The acceleration at some prompt of a runner as she ring the curve ~ above a track is 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?