Chapter 5, Problem 5A

Will a vector at $45°$ to the horizontal be larger or smaller than its horizontal and vertical components? By how much?

To determine

To identify: Whether a vector at given angle to horizontal is large or small than vertical and horizontal component. Also, state the amount of greater value.

Answer to Problem 5A

A vector at $45°$ to horizontal will be larger than the horizontal and vertical components of a vector.

It is 1.414 times greater than its components.

Explanation of Solution

Given:

The vector is at angle of $45°$ with the horizontal.

Formula used:

The horizontal and vertical components of vector Rat $\theta$ to the horizontal is,

  $\begin{array}{l}{R}_H=R\cos \theta \\ R_V=R\sin \theta \end{array}$

Calculation:

If a vector makes an angle of $45°$ with the horizontal then, horizontal and vertical components are:

  $\begin{array}{c}{R}_H=R\cos 45°\\ =R\times \frac{1}{\sqrt{2}}\\ =0.707R\end{array}$

And

  $\begin{array}{c}{R}_V=R\cos 45°\\ =R\times \frac{1}{\sqrt{2}}\\ =0.707R\end{array}$

From the above calculations, it is clear that vector R is greater than the horizontal and vertical components.

So,

  $R=\sqrt{2}\times R_H=\sqrt{2}\times R_V=1.414R_H\text{or 1}\text{.414}{R}_V$

It means the vector which is at $45°$ to the horizontal will be 1.414 times greater than its horizontal as well as vertical components.

Conclusion:

Therefore, a vector at $45°$ to horizontal will be larger than the horizontal and vertical components of a vector. It is 1.414 times greater than its components.