Question
If $$\overrightarrow a ,\,\overrightarrow b $$ are nonzero and noncollinear vectors then $$\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow i } \right]\overrightarrow i + \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow j } \right]\overrightarrow j + \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow k } \right]\overrightarrow k $$ is equal to :
A.
$$\overrightarrow a + \overrightarrow b $$
B.
$$\overrightarrow a \times \overrightarrow b $$
C.
$$\overrightarrow a - \overrightarrow b $$
D.
$$\overrightarrow b \times \overrightarrow a $$
Answer :
$$\overrightarrow a \times \overrightarrow b $$
Solution :
$$\eqalign{
& {\text{Let }}\overrightarrow a \times \overrightarrow b = x\overrightarrow i + y\overrightarrow j + z\overrightarrow k \cr
& \therefore \overrightarrow a \times \overrightarrow b .\overrightarrow i = x\,\,\overrightarrow a \times \overrightarrow b .\overrightarrow j = y\,\,\overrightarrow a \times \overrightarrow b .\overrightarrow k = z \cr
& \therefore \overrightarrow a \times \overrightarrow b = \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow i } \right]\overrightarrow i + \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow j } \right]\overrightarrow j + \left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow k } \right]\overrightarrow k \cr} $$