Q20DQQ - University Physics with Modern Physics

Question

Let A represent any nonzero vector. Why is $\raisebox{1ex}{$\overrightarrow{A}$}\!\left/ \!\raisebox{-1ex}{$A$}\right.$ a unit vector, and what is its direction? If $\theta$ is the angle that A makes with the +x-axis, explain why $\raisebox{1ex}{$\overrightarrow{A}$}\!\left/ \!\raisebox{-1ex}{$A$}\right.$  is called the direction cosine for that axis.

Step-by-Step Solution

Answer

Magnitude is 1 and its direction is the same as the vector, the quantity is called a unit vector.

Step 1: Unit Vector and its direction

The vector's various coordinates are presented here.

A vector having the magnitude of 1 and no units is called a unit vector. Its sole function is to point, or to describe a spatial direction.

 For many formulas involving vector components, unit vectors are a useful notation.

A unit vector dn pointing in the positive x-axis and a unit vector en pointing in the positive y-axis can be defined in a xy-coordinate system.

 We know that in coordinates, the x component of a vector comes first, followed by the y component. 

To calculate the angle of the vector with respect to the x axis, just multiply the y component by the x component and then take the inverse of that result.

This will tell you the vector's angle with the x axis.

Step 2: Unit Vector and direction cosine

To determine: The justification behind the unit vector's name, as well as its direction.

 The x-axis was given the moniker direction cosine for a purpose. 

As its magnitude is 1 and its direction is the same as the vector, the quantity is called a unit vector.

 Because it equals the angle formed by the vectors with the positive x-axis, the quantity is called direction cosine for x-axis. 

The quantity has a magnitude of one. As a result, it is referred to as a unit vector. Calculate the quantity's value.

 The vector's angle with the positive x-axis is Cosine direction. As a result, the quantity is known as x-axis direction cosine.