If a = i ^ + 2 k ^ , b = i ^ + j ^ + k ^ , c = 7 i ^ - 3 j ^ + 4 k ^ , r × b + b × c = 0 and r · a = 0 . Then r · c is equal to [2023]

Question

If $\vec{a} = \hat{i} + 2 \hat{k},$ $\vec{b} = \hat{i} + \hat{j} + \hat{k},$ $\vec{c} = 7 \hat{i} - 3 \hat{j} + 4 \hat{k},$ $\vec{r} \times \vec{b} + \vec{b} \times \vec{c} = \vec{0}$ and $\vec{r} \cdot \vec{a} = 0 .$ Then $\vec{r} \cdot \vec{c}$ is equal to [2023]

Smart Achievers · Accepted Answer

(1)

$\vec{a} = \hat{i} + 0 \hat{j} + 2 \hat{k}$

$\vec{b} = \hat{i} + \hat{j} + \hat{k};$ $\vec{c} = 7 \hat{i} - 3 \hat{j} + 4 \hat{k}$

$and \vec{r} \times \vec{b} + \vec{b} \times \vec{c} = 0 (given)$

$\Rightarrow \vec{r} \times \vec{b} - \vec{c} \times \vec{b} = 0 (∵ \vec{a} \times \vec{b} = - \vec{b} \times \vec{a})$

$\Rightarrow (\vec{r} - \vec{c}) \times \vec{b} = 0 \Rightarrow \vec{r} - \vec{c} = λ \vec{b} \Rightarrow \vec{r} = \vec{c} + λ \vec{b}$

$and \vec{r} \cdot \vec{a} = 0 is also given$

$\Rightarrow \vec{c} \cdot \vec{a} + λ \vec{b} \cdot \vec{a} = 0 or λ = - \frac{\vec{c} \cdot \vec{a}}{\vec{b} \cdot \vec{a}}$

$Now \vec{r} \cdot \vec{c} = (\vec{c} + λ \vec{b}) \cdot \vec{c}$

$= [\vec{c} - \frac{\vec{c} \cdot \vec{a}}{\vec{b} \cdot \vec{a}} \vec{b}] \cdot \vec{c} = | \vec{c} |^{2} - (\frac{\vec{c} \cdot \vec{a}}{\vec{b} \cdot \vec{a}}) (\vec{b} \cdot \vec{c})$

$= 74 - [\frac{15}{3}] 8 = 74 - 40 = 34$

Solution

Q.
If $\vec{a} = \hat{i} + 2 \hat{k},$ $\vec{b} = \hat{i} + \hat{j} + \hat{k},$ $\vec{c} = 7 \hat{i} - 3 \hat{j} + 4 \hat{k},$ $\vec{r} \times \vec{b} + \vec{b} \times \vec{c} = \vec{0}$ and $\vec{r} \cdot \vec{a} = 0 .$ Then $\vec{r} \cdot \vec{c}$ is equal to [2023]

1 34

2 36

3 32

4 30

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