Three Dimensional Geometry Test

Directions For Questions

Let $$A$$ be the given point whose position vectors with reference to origin $$O$$ be $$ \overrightarrow{a}$$ and $$ \overrightarrow{ON}= \overrightarrow{n}$$ Let $$P$$ be any point such that $$\overline{OP}= \overrightarrow{r}$$ lies on the plane & passes through $$A$$ and orthogonal to $$ON$$. Then for any point $$P$$ lies on the plane then.$$\overrightarrow{AP} \perp \overrightarrow{n}$$
$$\displaystyle \therefore \overrightarrow{AP}.\ \overrightarrow{n}=0$$
$$\displaystyle \Rightarrow \left ( \overrightarrow{OP}-\overrightarrow{OA} \right ).\overrightarrow{n}=0$$
$$\displaystyle \Rightarrow \overrightarrow{r}\cdot \overrightarrow{n}=\overrightarrow{a}.\overrightarrow{n}$$ $$($$Knows as scalar product form$$)$$
$$\displaystyle \Rightarrow \overrightarrow{r}\cdot \overrightarrow{n}=p,$$ where $$p$$ is the $$\perp$$er distance from origin to the plane.
On the bases of above information answer the following questions

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Vector equation of the plane passing through a point having position vector $$\displaystyle 2\hat{i}+3\hat{j}-4\hat{k}$$ and perpendicular to the vector $$\displaystyle 2\hat{i}-\hat{j}+2\hat{k}$$ is

Three Dimension...

Question 1

$$x + 2y + 3= 0$$

$$2x -y + 3 = 0$$

$$x + y +2z + 7= 0$$

$$4x + 3y =9 $$

Question 2

$$\displaystyle \overrightarrow{r}\cdot \left ( 2\hat{i}-\hat{j}+2\hat{k} \right )+7=0$$

$$\displaystyle \overrightarrow{r}\cdot \left ( 2\hat{i}-\hat{j}+2\hat{k} \right )=7$$

$$\displaystyle \overrightarrow{r}\cdot \left ( -3\hat{i}-2\hat{j}-3\hat{k} \right )=0$$

$$\displaystyle \overrightarrow{r}\cdot \left ( 2\hat{i}-\hat{j}+2\hat{k} \right )=9$$

Question 3

$$\displaystyle \dfrac{3}{5}$$

$$\displaystyle \frac{1}{5}$$

$$\displaystyle \frac{2}{3}$$

$$\displaystyle \frac{2}{5}$$

Question 4

$$a= 4, b= 6$$

$$a= 6, b= 4$$

$$a= 8, b= 2$$

$$a= 2, b= 8$$

Question 5

$$1$$

2

$$3$$

none of these

Question 6

$$\displaystyle (\bar{r}-\bar{a})\cdot (\bar{\alpha} \times \bar{\beta}) =0$$

$$\displaystyle (\bar{r}+\bar{a})(\bar{\alpha }\times \bar{\beta}) =0$$

$$\displaystyle (\bar{r}-\bar{a}) (\bar {a}\cdot \bar{b}) =0$$

none of these

Question 7

2

4

6

8

Question 8

$$\displaystyle < a_{1},a_{2},a_{3} > $$

$$\displaystyle <-a_{1},-a_{2},-a_{3}>$$

$$\displaystyle <\frac {a_{1}}{\left|\bar{v}\right|},\frac {a_{2}}{\left|\bar{v}\right|,}\frac {a_{3}}{\left|\bar{v}\right|},>$$

none of these

Question 9

$$\displaystyle \sin^{-1}\frac{1}{\sqrt{3}}$$

$$\displaystyle \cos^{-1}\frac{1}{\sqrt{6}}$$

$$\displaystyle \cos^{-1}\frac{1}{\sqrt{5}}$$

$$\displaystyle \cos^{-1}2/3$$

Question 10

$$x + y + z = 0$$

$$3x -2y -z =0$$

$$x -3y + 2z = 0$$

$$x + y + z = 42$$

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