Question
If $$\overrightarrow d $$ is a unit vector such that $$\overrightarrow d = \lambda \overrightarrow b \times \overrightarrow c + \mu \overrightarrow c \times \overrightarrow a + \nu \overrightarrow a \times \overrightarrow b $$ then $$\left| {\left( {\overrightarrow d .\overrightarrow a } \right)\left( {\overrightarrow b \times \overrightarrow c } \right) + \left( {\overrightarrow d .\overrightarrow b } \right)\left( {\overrightarrow c \times \overrightarrow a } \right) + \left( {\overrightarrow d .\overrightarrow c } \right)\left( {\overrightarrow a \times \overrightarrow b } \right)} \right|$$ is equal to :
A.
$$\left| {\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]} \right|$$
B.
$$1$$
C.
$$3\left| {\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]} \right|$$
D.
none of these
Answer :
$$\left| {\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]} \right|$$
Solution :
$$\eqalign{
& \overrightarrow d .\overrightarrow a = \lambda \left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow a } \right] + \mu \left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow a } \right] + \nu \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] = \lambda \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] \cr
& {\text{Similarly, }}\overrightarrow d .\overrightarrow b = \mu \left[ {\overrightarrow c \,\,\overrightarrow a \,\,\overrightarrow b } \right]{\text{ and }}\overrightarrow d .\overrightarrow c = \nu \left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right] \cr
& \therefore \,\lambda = \frac{{\overrightarrow d .\overrightarrow a }}{{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]}},\,\,\mu = \frac{{\overrightarrow d .\overrightarrow b }}{{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]}},\,\,\nu = \frac{{\overrightarrow d .\overrightarrow c }}{{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]}} \cr
& \therefore \,\overrightarrow d = \frac{{\overrightarrow d .\overrightarrow a }}{{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]}}\overrightarrow b \times \overrightarrow c + \frac{{\overrightarrow d .\overrightarrow b }}{{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]}}\overrightarrow c \times \overrightarrow a + \frac{{\overrightarrow d .\overrightarrow c }}{{\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]}}\overrightarrow a \times \overrightarrow b \cr
& \therefore {\text{ the expression}} = \left| {\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]\overrightarrow d } \right| = \left| {\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]} \right|\,\left| {\overrightarrow d } \right| = \left| {\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right]} \right| \cr} $$