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  • Need solution for RD sharma maths class 12 chapter 24 Vector or cross product exercise Fill in the blanks question 7

Need solution for RD sharma maths class 12 chapter 24 Vector or cross product exercise Fill in the blanks question 7

Answers (1)

 \lambda = 2

Hint: Use formula

\left | \vec{a} \times \vec{i }\right |^{2}=\left | \vec{a} \times \vec{i }\right |.\left | \vec{a} \times \vec{i }\right |

Given:

\left | \vec{r} \times \vec{i }\right |^{2}+\left | \vec{r} \times \vec{j}\right |^{2}+\left | \vec{r} \times \vec{k}\right |^{2}=\lambda |r|^{2}

Solving

\left | \vec{r} \times \vec{i }\right |^{2}=\left (\vec{r} \times \vec{i}\right ).\left ( \vec{r} \times \vec{i}\right )

=\begin{vmatrix} \hat{r}.\hat{r} & \hat{r}.\hat{i} \\ \hat{i}.\hat{r} & \hat{i}.\hat{i} \end{vmatrix}\\ =\begin{vmatrix} \hat{r}.\hat{r} & \hat{r}.\hat{i} \\ \hat{i}.\hat{r} & 1 \end{vmatrix}=\begin{vmatrix} \hat{r}.\hat{r} & \hat{r}.\hat{i} \\ \hat{i}.\hat{r} & \hat{i}.\hat{i} \end{vmatrix}\\ =\begin{vmatrix} \hat{r}.\hat{r} & \hat{r}.\hat{i} \\ \hat{i}.\hat{r} & 1 \end{vmatrix}

=|\vec{r}|^{2}-(\vec{r}.\vec{i})^{2}............................(1)

Similarly

|\vec{r} \times \vec{j}|^{2}=|\vec{r}|^{2}-(\vec{r} \cdot \vec{j})^{2} ................(2)

|\vec{r} \times \vec{k}|^{2}=|\vec{r}|^{2}-(\vec{r} \cdot \vec{k})^{2}.................(3)

Adding (1), (2) & (3)

we get,

\begin{aligned} &|\vec{r} \times \vec{i}|^{2}+|\vec{r} \times \vec{j}|^{2}+|\vec{r} \times \vec{k}|^{2} \\ &=\left[|\vec{r}|^{2}-|\vec{r} \cdot \vec{i}|^{2}\right]+\left[|\vec{r}|^{2}-|\vec{r} \cdot \vec{j}|^{2}\right]+\left[|\vec{r}|^{2}-|\vec{r} \cdot \vec{k}|^{2}\right] \\ \end{aligned}

\begin{aligned} &\vec{r} .\hat{i}=(\hat{i}+\hat{j}+\hat{k}) \cdot(\hat{i}+0 \hat{j}+0 \hat{k}) \\ &=(\hat{i})^{2} \\ \end{aligned}

\begin{aligned} &\vec{r} \cdot \vec{j}=(\hat{i}+\hat{j}+\hat{k}) \cdot(0 \hat{i}+\hat{j}+0 \hat{k}) \\ &=(\hat{j})^{2} \\ \end{aligned}

\begin{aligned} &\vec{r} \cdot \vec{k}=(\hat{i}+\hat{j}+\hat{k}) \cdot(0 \hat{i}+0 \hat{j}+\hat{k}) \\ &=(\hat{k})^{2} \\ \end{aligned}

\begin{aligned} &\Rightarrow\left[|\vec{r}|^{2}+|\vec{r}|^{2}+|\vec{r}|^{2}\right]-\left[|\vec{r} \cdot \vec{i}|^{2}+|\vec{r} \cdot \vec{j}|^{2}+|\vec{r} \vec{k}|^{2}\right] \\ \end{aligned}

\begin{aligned} &=3|\vec{r}|^{2}-\left[\left|\hat{i}^{2}+\hat{j}^{2}+\hat{k}^{2}\right|\right] \\ &=3|\vec{r}|^{2}-|\vec{r}|^{2} \\ &=2|\vec{r}|^{2} \\ \end{aligned}

We have been given,

 \begin{aligned}&|\vec{r} \times \vec{i}|^{2}+|\vec{r} \times \vec{j}|^{2}+|\vec{r} \times \vec{k}|^{2}=\lambda|\vec{r}|^{2} \\ &\lambda=2 \end{aligned}

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