If $$\overrightarrow a \,.\,\overrightarrow b = 1,\,\overrightarrow b \,.\,\overrightarrow c = 2$$ and $\overrightarrow c \,.\,\overrightarrow a = 3$, then the value of $$\left[ {\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right),\,\overrightarrow b \times \left( {\overrightarrow c \times \overrightarrow a } \right),\,\overrightarrow c \times \left( {\overrightarrow b \times \overrightarrow a } \right)} \right]$$ is :
Solution
<p>$\because$ $$\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right) = 3\overrightarrow b - \overrightarrow c = \overrightarrow u $$</p>
<p>$$\overrightarrow b \times \left( {\overrightarrow c \times \overrightarrow a } \right) = \overrightarrow c - 2\overrightarrow a = \overrightarrow v $$</p>
<p>$$\overrightarrow c \times \left( {\overrightarrow b \times \overrightarrow a } \right) = 3\overrightarrow b - 2\overrightarrow a = \overrightarrow w $$</p>
<p>$\therefore$ $\overrightarrow u + \overrightarrow v = \overrightarrow w$</p>
<p>So, vectors $\overrightarrow u$, $\overrightarrow v$ and $\overrightarrow w$ are coplanar, hence their Scalar triple product will be zero.</p>
About this question
Subject: Mathematics · Chapter: Vector Algebra · Topic: Types of Vectors
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