Let $\overrightarrow a = \widehat i + \widehat j + 2\widehat k$ and $\overrightarrow b = - \widehat i + 2\widehat j + 3\widehat k$. Then the vector product $$\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\left( {\overrightarrow a \times \left( {\left( {\overrightarrow a - \overrightarrow b } \right) \times \overrightarrow b } \right)} \right) \times \overrightarrow b } \right)$$ is equal to :
Solution
$\overrightarrow a = \widehat i + \widehat j + 2\widehat k$<br><br>$\overrightarrow b = - \widehat i + 2\widehat j + 3\widehat k$<br><br>$$\overrightarrow a + \overrightarrow b = 3\widehat j + 5\widehat k;\overrightarrow a.\overrightarrow b = - 1 + 2 + 6 = 7$$<br><br>$$\left( {\left( {\overrightarrow a \times \left( {\left( {\overrightarrow a - \overrightarrow b } \right) \times \overrightarrow b } \right)} \right) \times \overrightarrow b } \right)$$<br><br>$$\left( {\left( {\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow b - \overrightarrow b \times \overrightarrow b } \right)} \right) \times \overrightarrow b } \right)$$<br><br>$$\left( {\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow b - 0} \right)} \right) \times \overrightarrow b $$<br><br>$$\left( {\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow b } \right)} \right) \times \overrightarrow b $$<br><br>$$\left( {\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow a - \left( {\overrightarrow a .\overrightarrow a } \right)\overrightarrow b } \right) \times \overrightarrow b $$<br><br>$$\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow a \times \overrightarrow b - \left( {\overrightarrow a .\overrightarrow a } \right)\left( {\overrightarrow b \times \overrightarrow b } \right)$$<br><br>$$\left( {\overrightarrow a .\overrightarrow b } \right)\left( {\overrightarrow a \times \overrightarrow b } \right)$$<br><br>$$\overrightarrow a \times \overrightarrow b = \left| {\matrix{
i & j & k \cr
1 & 1 & 2 \cr
{ - 1} & 2 & 3 \cr
} } \right| = - \widehat i - 5\widehat j + 3\widehat k$$<br><br>$\therefore$ $7\left( { - \widehat i - 5\widehat j + 3\widehat k} \right)$<br><br>$$\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {7\left( { - \widehat i - 5\widehat j + 3\widehat k} \right)} \right)$$<br><br>$$7\left( {0\widehat i + 3\widehat j + 5\widehat k} \right) \times \left( { - \widehat i - 5\widehat j + 3\widehat k} \right)$$<br><br>$$\left| {\matrix{
{\widehat i} & {\widehat j} & {\widehat k} \cr
0 & 3 & 5 \cr
{ - 1} & { - 5} & 3 \cr
} } \right|$$<br><br>$\Rightarrow 34\widehat i - (5)\widehat j + (3\widehat k)$<br><br>$\Rightarrow 34\widehat i - 5\widehat j + 3\widehat k$<br><br>$\therefore$
$$7\left( {0\widehat i + 3\widehat j + 5\widehat k} \right) \times \left( { - \widehat i - 5\widehat j + 3\widehat k} \right)$$
<br><br>= $7(34\widehat i - 5\widehat j + 3\widehat k)$
About this question
Subject: Mathematics · Chapter: Vector Algebra · Topic: Types of Vectors
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