Chapter 6 - Section 6.6 - Vectors - Exercise Set - Page 782: 47

$\mathbf{v}=3\sqrt{3}\mathbf{i}+3\mathbf{j}$

Work Step by Step

Consider the following vector $\mathbf{v}$, $\mathbf{v}=a\mathbf{i}+b\mathbf{j}$ (I) The components $a$ and $b$ can be expressed in terms of the magnitude of vector $\mathbf{v}$ and direction angle $\theta$ as, \begin{align} & a=\parallel \mathbf{v}\parallel \cos \theta \\ & b=\parallel \mathbf{v}\parallel \sin \theta \\ \end{align} Substitute the value of $a,b$ and $\theta$ in equation (I) to get, \begin{align} & \mathbf{v}=\parallel v\parallel \cos {{30}^{\circ }}\mathbf{i}+\parallel v\parallel \sin {{30}^{\circ }}\mathbf{j} \\ & =\left( 6\times \frac{\sqrt{3}}{2} \right)\mathbf{i}+\left( 6\times \frac{1}{2} \right)\mathbf{j} \\ & =3\sqrt{3}\mathbf{i}+3\mathbf{j} \end{align}

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