Question

What is the magnitude $\overrightarrow{\mathbf{a}}\mathbf{\times }\mathbf{(}\overrightarrow{\mathbf{b}}\mathbf{\times }\overrightarrow{\mathbf{a}\mathbf{)}}$of if a = 3.90 , b = 2.70and the angle between the two vectors is localid="1656307928946" $\mathbf{63}\mathbf{.}{\mathbf{0}}^{\mathbf{°}}$?

Short answer

The magnitude is36.6

Step-by-step solution

To understand the concept

This problem is based on the vector product. When two vectors are multiplied by the sine of the angle between them, the magnitude of the vector product can be found.

The vector product of two vectors $\overrightarrow{a}$and $\overrightarrow{b}$is given by

$\overrightarrow{c}=(\overrightarrow{b}\times \overrightarrow{a})$ (1)

And the magnitude is given by

$$\begin{gathered} \mathrm{c}=\mathrm{ab}\mathrm{sin\theta }\left(2\right) \\ \mathrm{Given} \\ \mathrm{a}=3.90 \\ \mathrm{b}=2.70 \\ \mathrm{Angle}\mathrm{between}\mathrm{these}\mathrm{two}\mathrm{vectors}\mathrm{is}\mathrm{\theta }=36.6 \end{gathered}$$

To find the magnitude ofa→×(b→×a)→

Here, vector is perpendicular to vector$\overrightarrow{a}$Thus the magnitude of role="math" localid="1656307375446" $(\overrightarrow{a}\times \overrightarrow{c})$is given by,

$ac\sin \left(90\right)=ac$

Using equation (i) and (ii), the magnitude of $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{a})$is

$\left|\overrightarrow{\mathrm{a}}\times (\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{a})}\right|=\mathrm{ac}={\mathrm{a}}^2\mathrm{b}\mathrm{sin\theta }$ (3)

Substituting the given values of a and b in equation (iii),

$\left|\overrightarrow{\mathrm{a}}\times (\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{a})}\right|={(3.90)}^2(2.70)\sin \left(63\right)=36.6$