Question

Components of a Vector

Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors$i$and $\mathit{j}$

$\left|\mathbf{v}\right|=50,\theta ={120}^{°}$

Short answer

The required representation of vector is$v=-25i+25\sqrt{3}j$and horizontal component is $-25$and vertical component is $25\sqrt{3}$

Step-by-step solution

Step 1. Given information

A modulus of vector and an angle are given as $\left|\mathbf{v}\right|=50,\theta ={120}^{°}$

Step 2. Concept used

When a vector, v , has its magnitude as$\left|v\right|$and direction as $\theta$, then

$v=a_{1}i+a_{2}j$where.

So we can write vector as-$\begin{array}{l}{a}_1=\left|v\right|\cos \theta \\ a_2=\left|v\right|\sin \theta \end{array}$

$v=\left|v\right|\cos \theta i+\left|v\right|\sin \theta j$

Use the formulas to get the required answers.

Step 3. Calculation

Substitute the values of $u$and $v$in the required terms,

$\begin{array}{l}{a}_1=\left|v\right|\cos \theta \\ =50\cos {120}^{°}\\ =50\left(\frac{-1}{2}\right)\\ =-25\end{array}$

And

$\begin{array}{l}{a}_2=\left|v\right|\sin \theta \\ =50\sin {120}^{°}\\ =50\left(\frac{\sqrt{3}}{2}\right)\\ =25\sqrt{3}\end{array}$

So, the vector is-

$v=-25i+25\sqrt{3}j$