Question

(a) A unit vector lies in the xy plane, at an angle of $160°$from the +x axis with a positive y component. What is the unit vector? (It helps to draw a diagram). (b) A string runs up and to the left in the xy plane, making an angle of $40°$to the vertical. Determine the unit that points along the string.

Short answer

(a) The unit vector is .

(b) The unit vector is .

Step-by-step solution

Definition of unit vector

A vector is a quantity that has both a magnitude and a direction associated with it. A unit vector is a vector with a magnitude of one. Unit vectors form the basis of vector calculus.

Unit vector given by the direction cosines is.

Finding unit vector

(a)

Draw the diagram according to problem statement.

Write the angles from the figure.

${\theta }_x=160°$

${\theta }_y=70°$

${\theta }_z=90°$

Substitute these values into the formula of unit vector to obtain the required unit vector .

Thus, the required unit vector is .

Finding unit vector

(b)

Draw the diagram according to problem statement.

Write the angles from the figure.

$$\begin{gathered} {\theta }_x=130° \\ {\theta }_y=40° \\ {\theta }_z=90° \end{gathered}$$

Substitute these values into the formula of unit vector to obtain the required unit vector.

Thus, the required unit vector is.