Question

A string runs up and to the left in thex⁢yplane, making an angle of 39 degrees to the vertical. Determine the unit vector that points along the string.

Answer 1

Answer:

$\vec v=-0.63\hat i+0.78\hat j$

Explanation:

Vectors

It's an element that has both magnitude and direction. It can be geometrically represented as a directed line segment whose length is the magnitude and the arrow points in its direction. There are several ways to mathematically represent a vector. The most commonly used are in the polar and rectangular coordinates. In the polar form, a vector has a magnitude and an angle measured respect to the horizontal right direction. It's written as . The rectangular representation has the two coordinates (x,y) measured in the known plane xy. The conversion between both systems is

$x=rcos\theta$

$y=rsin\theta$

A representation of the vector is shown in the figure below. The angle of $39^o$ is measured respect to the vertical, we must add $90^o$ to make it compliant with the standard expressions of vectors. We have r=1, $\theta=39^o+90^o=129^o$

$x=(1)cos129^o=-0.63$

$y=(1)sin129^o=0.78$

The unit vector is

$\vec v=<-0.63,0.78>$

Or equivalently

$\boxed {\vec v=-0.63\hat i+0.78\hat j}$