Download App

Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry3 Marks
Question
Show that the points $(\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}})$ and $3(\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}})$ are equidistant from the plane $\vec{\text{r}}\cdot(5\hat{\text{i}}+2\hat{\text{j}}-7\hat{\text{k}})+9=0$ and lies on opposite side of it.
✓

Answer

To show that these given points $(\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}})$ and $3(\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}})$ are equidistant from the plane
$\vec{\text{r}}\cdot(5\hat{\text{i}}+2\hat{\text{j}}-7\hat{\text{k}})+9=0,$ we first find out the mid-point of the points which is $(2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}).$
On substituting $\vec{\text{r}}$ by the mid-point in plane, we get
$\text{L.H.S.}=(2\hat{\text{i}}+\hat{\text{j}}+3\hat{\text{k}})\cdot(5\hat{\text{i}}+\hat{\text{j}}-7\hat{\text{k}})+9$
$=10+2-21+9=0$
$=\text{R.H.S.}$
Hence, the two points lie on opposite sides of the plane are equidistant from the plane.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Three Dimensional Geometry→Three Dimensional Geometry — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Solve the following $\text{LPP}$ by graphical method:
Minimize $Z = 20x + 10y$
Subject to $x+2 y \leq 40 , 3 x+y \geq 30 , 4 x+3 y \geq 60$ and $ x, y \geq 0$
→
Evaluate $\sin\Big(\tan^{-1}\frac{3}{4}\Big).$
→
Find a point on the curve $y = (x – 2)^2 $ at which the tangent is parallel to the chord joining the points $(2, 0)$ and $(4, 4).$
→
Classify the following functions as injection, surjection or bijection:
$f : N \rightarrow N$ given by $f(x) = x^3$​​​​​​​
→
Check whether the relation $R$ on $R$ defined by $R = \{(a, b): a \leq b^3\}$ is reflexive, symmetric or transitive.
→
If $\text{A}=\begin{bmatrix}1&2\\4&1\end{bmatrix},$ then find $A^2 + 2A + 7I$.
→
If the function f: R$\rightarrow$R be given by $f(x) = x^2 + 2$ and g: R$\rightarrow$R be given by g(x) $ = \frac{\text{x}}{\text{x} - 1 },\text{x}\neq1 ,$ find fog and gof and hence find fog (2) and gof (–3).
→
Write a value of $\int(\text{e}^{\text{x}\log_\text{e}\text{a}}+\text{e}^{\text{a}\log_\text{e}\text{x}})\text{dx}$
→
If $\text{A}=\text{diag}\begin{pmatrix}2&-5&9\end{pmatrix},\text{ B}=\text{diag}\begin{pmatrix}1&1&-4\end{pmatrix}$ and $\text{C}=\text{diag}\begin{pmatrix}-6&3&4\end{pmatrix},$ find.
$2\text{A}+3\text{B}-5\text{C}$
→
Evaluate $\int \frac{\sin (x-a)}{\sin (x+a)} d x$
→