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Let the vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}$ are given $\mathbf{a}_1 \hat{\mathbf{i}}+\mathbf{a}_2 \hat{\mathbf{j}}+\mathbf{a}_3 \hat{\mathbf{k}}$, $\mathbf{b}_1 \hat{\mathbf{i}}+\mathbf{b}_2 \hat{\mathbf{j}}+\mathbf{b}_3 \hat{\mathbf{k}}$ and $\mathbf{c}_1 \hat{\mathbf{i}}+\mathbf{c}_2 \hat{\mathbf{j}}+\mathbf{c}_3 \hat{\mathbf{k}}$. Then show that $\overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}})=\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{c}}$
MathematicsVector Algebra
Solution:
2901 Upvotes Verified Answer
$\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\left(\mathrm{b}_1 \hat{\mathrm{i}}+\mathrm{b}_2 \hat{\mathrm{j}}+\mathrm{b}_3 \hat{\mathrm{k}}\right)+\left(\mathrm{c}_1 \hat{\mathrm{i}}+\mathrm{c}_2 \hat{\mathrm{j}}+\mathrm{c}_3 \hat{\mathrm{k}}\right)$
$=\left(\mathrm{b}_1+\mathrm{c}_1\right) \hat{\mathrm{i}}+\left(\mathrm{b}_2+\mathrm{c}_2\right) \hat{\mathrm{j}}+\left(\mathrm{b}_3+\mathrm{c}_3\right) \hat{\mathrm{k}}$
L.H.S. $=\vec{a} \times(\vec{b}+\vec{c})=\left|\begin{array}{ccc}i & j & k \\ a_1 & a_2 & a_3 \\ b_1+c_1 & b_2+c_2 & b_3+c_3\end{array}\right|$
$=\left[\left(\mathrm{a}_2 \mathrm{~b}_3-\mathrm{a}_3 \mathrm{~b}_2\right)+\left(\mathrm{a}_2 \mathrm{c}_3-\mathrm{a}_3 \mathrm{c}_2\right)\right]$
$\hat{i}-\left[\left(a_1 b_3-a_3 b_1\right)+\left(a_1 c_3-c_3 c_1\right) \hat{j}\right]$
$\left.+\left[a_1 b_2-a_2 b_1\right)+\left(a_1 c_2-a_2 c_1\right) \hat{k}\right]$
R.H.S $=\vec{a} \times \vec{b}+\vec{a} \times \vec{c}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\end{array}\right|+\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ c_1 & c_2 & c_3\end{array}\right|$
$=\left[\left(a_2 b_3-a_3 b_2\right) \hat{i}-\left(a_1 b_3-a_3 b_1\right)\right.$
$\left.\hat{j}+\left(a_1 b_2-a_2 b_1\right) \hat{k}\right]+\left[\left(a_2 c_3-a_3 c_2\right)\right.$
$\left.\hat{i}-\left(a_1 c_3-a_3 c_1\right) \hat{j}+\left(a_1 c_2-a_2 c_1\right) \hat{k}\right]$
from (i) and (ii), we conclude that
$\vec{a} \times(\vec{b}+\vec{c})=\vec{a} \times \vec{b}+\vec{a} \times \vec{c}$

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