Question

Magnitude and Direction of a Vector

Find the magnitude and direction (in degrees) of the vector.

$\mathbf{v}=⟨3,4⟩$

Short answer

The magnitude of the vector is$5$and the direction(in degrees) is $\theta ={53}^{°}$

Step-by-step solution

Step 1. Given information

A point is given as $\mathbf{v}=⟨3,4⟩$

Step 2. Concept used

The formula used to calculate the length or magnitude of a vector is-

$\left|v\right|=\sqrt{a_1^2+a_2^2}$

And for the direction or angle, we use-

$\tan \theta =\frac{a_2}{a_1}$

Use the formulas to get the required answers.

Step 3. Calculation

The magnitude of vector is-

$\begin{array}{l}\left|v\right|=\sqrt{3^2+4^2}\\ =\sqrt{9+16}\\ =\sqrt{25}\\ =5\end{array}$

And the direction is-

$\begin{array}{l}\tan \theta =\frac{4}{3}\\ \theta ={\tan }^{-1}\left(\frac{4}{3}\right)\\ ={53}^{°}\end{array}$