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Explain solution RD Sharma class 12 Chapter 23 Scalar or Dot Products Exercise Very Short Answer question 17

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ANSWER: $3,-4,12$

GIVEN: $\vec{r}=3 \hat{i}-4 \hat{j}+12 \hat{k}$

HINTS: Find the projection on each coordinate axis one by one.

SOLUTION:We know that,

Component along x-axis= $\hat{i}$

Component along y-axis= $\hat{j}$

Component along z-axis= $\hat{k}$

Now projection of : $\vec{r}$ on x axis = $\frac{\vec{r\cdot \hat{i}}}{\left | \hat{i} \right |}$

$\begin{aligned} &=\frac{(3 \hat{i}-4 \hat{j}+12 \hat{k}) \hat{.}}{\sqrt{1}} \\ & \end{aligned}$

$=\frac{3-0+0}{1}=3\qquad[\because \hat{i} \times \hat{i}=1, \hat{i} \times \hat{j}=0, \hat{i} \times \hat{k}=0]$

projection of $\vec{r}$ on y axis $\frac{\vec{r\cdot \hat{j}}}{\left | \hat{j} \right |}$

$\begin{aligned} &=\frac{(\hat{i}-4 \hat{j}+12 \hat{k}) \hat{. j}}{\sqrt{1}} \\ & \end{aligned}$

$=\frac{0-4+0}{1}=-4\qquad[\because \hat{i} \times \hat{i}=0, \hat{j} \times \hat{j}=1, \hat{j} \times \hat{k}=0]$

projection of $\vec{r}$ on y axis $\frac{\vec{r\cdot \hat{k}}}{\left | \hat{k} \right |}$

$\begin{aligned} &=\frac{(3 \hat{i}-4 \hat{j}+12 \hat{k}) \cdot \hat{k}}{\sqrt{1}} \\ & \end{aligned}$

$=\frac{0-0+12}{1}=12\qquad[\because \hat{i} \times \hat{k}=0, \hat{i} \times \hat{k}=0, \hat{k} \times \hat{k}=1]$

Hence ,required answer is $3,-4,12$

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