Question

If $\overrightarrow a ,\overrightarrow b ,\overrightarrow c$ form a left-handed orthogonal system and$\overrightarrow a .\overrightarrow a = 4,\overrightarrow b .\overrightarrow b = 9,\overrightarrow {c.} \overrightarrow c = 16$, then find the value of$\left[ {\overrightarrow a ,\overrightarrow b ,\overrightarrow c } \right]$
A. $24$
B. $- 24$
C. $12$
D. $- 12$

Answer 1

Initially, we will find the magnitude of each vector. Then using some formulas which are mentioned below, we will find our required answer.

Used formula: $\overrightarrow a .\overrightarrow a = {\left| {\overrightarrow a } \right|^2}$
$\overrightarrow b \times \overrightarrow c = \left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|\sin \theta$
$\left[ {\overrightarrow a ,\overrightarrow b ,\overrightarrow c } \right] = \overrightarrow a .(\overrightarrow b \times \overrightarrow c )$

Complete answer:.It is given that, $\overrightarrow a ,\overrightarrow b ,\overrightarrow c$ form a left-handed orthogonal system.
Also provided that, $\overrightarrow a .\overrightarrow a = 4,\overrightarrow b .\overrightarrow b = 9,\overrightarrow {c.} \overrightarrow c = 16$
We know that,
$\overrightarrow a .\overrightarrow a = {\left| {\overrightarrow a } \right|^2}$,
So, according to the problem using the values given we get, $\overrightarrow a .\overrightarrow a = {\left| {\overrightarrow a } \right|^2} = 4$
So, we have, ${\left| {\overrightarrow a } \right|} = 2$
Similarly, we will find ${\left| {\overrightarrow b } \right|} = 3$, ${\left| {\overrightarrow c } \right|} = 4$
Now we know that $\left[ {\overrightarrow a ,\overrightarrow b ,\overrightarrow c } \right]$ is given by the formula,
$\left[ {\overrightarrow a ,\overrightarrow b ,\overrightarrow c } \right] = \overrightarrow a .(\overrightarrow b \times \overrightarrow c )$
Here, the angle between the vectors $\overrightarrow a ,\overrightarrow b \times \overrightarrow c$ is ${180^ \circ }$. Since, $\overrightarrow b \times \overrightarrow c$ is exactly opposite to the vector $\overrightarrow a ,$
On simplifying using the angle mentioned above we get,
$\overrightarrow a .(\overrightarrow b \times \overrightarrow c ) = \left| {\overrightarrow a } \right|.\left| {\overrightarrow b \times \overrightarrow c } \right|\cos {180^ \circ }$
We know the trigonometric values of $\cos \theta$then we get, $\cos {180^ \circ } = - 1$
Substitute the value into the above expression we get,
$\overrightarrow a .(\overrightarrow b \times \overrightarrow c ) = - \left| {\overrightarrow a } \right|.\left| {\overrightarrow b \times \overrightarrow c } \right|$ …. (1)
Now we will consider the value of$\left| {\overrightarrow b \times \overrightarrow c } \right|$.
$\left| {\overrightarrow b \times \overrightarrow c } \right| = \left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|\sin {90^ \circ }$
Since, the angle between the vectors $\overrightarrow b$ and $\overrightarrow c$ is ${90^ \circ }$the value of $\theta$ is replaced by${90^ \circ }$ .
Substitute this value at the expression (1) we get,
$\overrightarrow a .(\overrightarrow b \times \overrightarrow c ) = - \left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|\sin {90^ \circ }$
Applying the trigonometric value of sine function we get,
$\overrightarrow a .(\overrightarrow b \times \overrightarrow c ) = - \left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|$
Now let us put the values of $\left| {\overrightarrow a } \right| = 2$,$\left| {\overrightarrow b } \right| = 3$, $\left| {\overrightarrow c } \right| = 4$ in the above equation, we get,
$\overrightarrow a .(\overrightarrow b \times \overrightarrow c ) = - 2 \times 3 \times 4 = - 24$
Hence,
$\overrightarrow a .(\overrightarrow b \times \overrightarrow c ) = - 24$
That is the value of $\left[ {\overrightarrow a ,\overrightarrow b ,\overrightarrow c } \right]$is$- 24$

Therefore, the correct option is (B)$- 24$.

Note: Let us consider the two vectors $\overrightarrow b$ and $\overrightarrow c$.
Then, $\overrightarrow b \times \overrightarrow c = \left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|\sin \theta$
Since, the given system is left-handed, orthogonal the angle between vectors $\overrightarrow b$ and \overrightarrow c $$$${90^ \circ }.
Again,
$\left[ {\overrightarrow a ,\overrightarrow b ,\overrightarrow c } \right] = \overrightarrow a .(\overrightarrow b \times \overrightarrow c )$
$\left[ {\overrightarrow a ,\overrightarrow b ,\overrightarrow c } \right]$is defined as the box product of the given vectors, the box product is nothing but the combination of dot product with cross product.