What is the value of linear velocity if $\vec{\omega} = 3\hat{i} - 4\hat{j} + \hat{k}$ and $\vec{r} = 5\hat{i}

Question

What is the value of linear velocity if $\vec{\omega} = 3\hat{i} - 4\hat{j} + \hat{k}$ and $\vec{r} = 5\hat{i} - 6\hat{j} + 6\hat{k}$:

Accepted Answer

1.  **Formula:** The relationship between linear velocity ($\mathbf{v}$), angular velocity ($\boldsymbol{\omega}$), and the position vector ($\mathbf{r}$) is given by the vector product (cross product): $\mathbf{v} = \boldsymbol{\omega} 	imes \mathbf{r}$ .
    Note: The order is important; $\mathbf{r} 	imes \boldsymbol{\omega}$ would give the opposite sign.
2.  **Calculation:** Perform the cross product using the determinant method:
    $$ \mathbf{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 3 & -4 & 1 \ 5 & -6 & 6 \end{vmatrix} $$
    $$ \mathbf{v} = \hat{i}((-4)(6) - (1)(-6)) - \hat{j}((3)(6) - (1)(5)) + \hat{k}((3)(-6) - (-4)(5)) $$
    $$ \mathbf{v} = \hat{i}(-24 + 6) - \hat{j}(18 - 5) + \hat{k}(-18 + 20) $$
    $$ \mathbf{v} = -18\hat{i} - 13\hat{j} + 2\hat{k} $$ .

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