Download App

Scalar Triple Product — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsScalar Triple Product5 Marks
Question
If four points A, B, C and D with position vectors $4\hat{\text{i}}+3\hat{\text{j}},5\hat{\text{i}}+\text{x}\hat{\text{j}}+7\hat{\text{k}},5\hat{\text{i}}+3\hat{\text{j}}$ and $7\hat{\text{i}}+6\hat{\text{j}}+\hat{\text{k}}$ respectively are coplanar, then find the value of x.
✓

Answer

Let $\vec{\text{OA}}=4\vec{\text{i}}+3\vec{\text{j}}+3\vec{\text{k}},\vec{\text{OB}}=5\hat{\text{i}}+\text{x}\hat{\text{j}}+7\hat{\text{k}},\vec{\text{OC}}=5\hat{\text{i}}+3\hat{\text{j}}$ and $7\hat{\text{i}}+6\hat{\text{j}}+\hat{\text{k}}.$
$\therefore\vec{\text{AB}}=(5\hat{\text{i}}+\text{x}\hat{\text{j}}+7\hat{\text{k}})-(4\hat{\text{i}}+3\hat{\text{j}}+3\hat{\text{k}})\\=\hat{\text{i}}+(\text{x}-3)\hat{\text{j}}+4\hat{\text{k}}$
$\vec{\text{AC}}=(5\hat{\text{i}}+3\hat{\text{j}})-(4\hat{\text{i}}+3\hat{\text{j}}+3\hat{\text{k}})\\=\hat{\text{i}}-3\hat{\text{k}}$
$\vec{\text{AD}}=(7\hat{\text{i}}+6\hat{\text{j}}+\hat{\text{k}})-(4\hat{\text{i}}+3\hat{\text{j}}+3\hat{\text{k}})\\=3\hat{\text{i}}+3\hat{\text{j}}-2\hat{\text{k}}$
Since the given four points are coplanar, so the vectors $\vec{\text{AB}},\vec{\text{AC}}$ and $\vec{\text{AD}}$ are also coplanar.
$\therefore\big[\vec{\text{AB}}\vec{\text{ AC }}\vec{\text{AD}}\big]=0$
$\begin{vmatrix}1&\text{x}-3&4\\1&0&-3\\3&3&-2 \end{vmatrix}=0$
$\Rightarrow 1(0+9)-(\text{x}-3)(-2+9)+4(3-0)=0$
$\Rightarrow 9-7\text{x}+21+12=0$
$\Rightarrow7\text{x}=42$
$\Rightarrow \text{x}=6$
Thus, the value of x is 6.

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 "Solve the Following Question.(5 Marks)" from Scalar Triple Product→Scalar Triple Product — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the point on the curve $y = 3x^2 + 4$ at which the tangent is perpendicular to the line whose slop is $-\frac{1}{6}$
→
A unit vector $\vec{\text{a}}$ makes angles $\frac{\pi}{4}$ and $\frac{\pi}{3}$ with $\hat{\text{i}}$ and $\hat{\text{j}}$ respectively and an acute angle $\theta$ with $\hat{\text{k}}$. find the angle $\theta$ and components of $\vec{\text{a}}$ .
→
If $\log y=\log (\sin x)-x^2$, show that $\frac{d^2 y}{d x^2}+4 x \frac{d y}{d x}+\left(4 x^2+3\right) y=0$
→
Let $A = {-1, 0, 1}$ and $f = {(x, x^2): x \in A}$. Show that $f : A \rightarrow A$ is neither one-one nor onto.
→
Solve the following systems of linear equations by cramer's rule:
$6x + y - 3z = 5,$
$x + 3y - 2z = 5,$
$2x + y + 4z = 8$
→
Consider $f : \{1, 2, 3\} \rightarrow \{a, b, c\}$ and $g : \{a, b, c\} \rightarrow \{$apple, ball, cat$\}$ defined as $f(1) = a, f(2) = b, f(3) = c, g(a) =$ apple, $g(b) =$ ball and $g(c) =$ cat. Show that $f, g $ and gof are invertible. Find $f^{-1}, g^{-1} $ and $gof^{-1} $ and show that $(gof)^{-1} = f^{-1}og^{-1}.$
→
Find the intervals in which the following functions are increasing or decreasing.
$f(x) = x^4 - 4x^3 - 4x^2 + 15$
→
Water is being poured into a vessel in the form of an inverted right circular cone of semi vertical angle $45^{\circ} \mathrm{c}$ in such a way that the rate of change of volume at any moment is proporational to the area of the curved surfaces which is wet at that moment. Initially, the vessel is full to a height of $2 \mathrm{cms}$. And after 2 seconds the height becomes $10 \mathrm{~cm}$. Show that after 3.5 seconds from that start, the height of water will be $16 \mathrm{cms}$.
→
Find the equation of the lines joining the following pairs of vertices and then find the shortest distance between the lines(1) (0, 0, 0) and (1, 0, 2)
(2) (1, 3, 0) and (0, 3, 0)
→
Solve the following systems of linear equations by cramer's rule:
9x + 5y = 10,
3x - 2y = 8
→