Download App

Scalar Triple Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsScalar Triple Product3 Marks
Question
Find the value of $\lambda$ so that the following vectors are coplanar:
$\vec{\text{a}}=2\hat{\text{i}}-\hat{\text{j}}++\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}},\vec{\text{c}}=\lambda\hat{\text{i}}+\lambda\hat{\text{j}}+5\hat{\text{k}}$
✓

Answer

Given:
$\vec{\text{a}}=2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$
$\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}$
$\vec{\text{c}}=\lambda\hat{\text{i}}+\lambda\hat{\text{j}}+5\hat{\text{k}}$
We know that vectors $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are coplanar iff $\Big[\vec{\text{a}}\ \vec{\text{b}}\ \vec{\text{c}}\Big]=0.$
It is given that $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are coplanar.
$\therefore\Big[\vec{\text{a}}\ \vec{\text{b}}\ \vec{\text{c}}\Big]=0$
$\Rightarrow\begin{vmatrix}2&-1&1\\1&2&-3\\\lambda&\lambda&5 \end{vmatrix}=0$
$\Rightarrow2(10+3\lambda)+1(5+3\lambda)+1(\lambda-2\lambda)=0$
$\Rightarrow 8\lambda +25=0$
$=-\frac{25}{8}$

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 Scalar Triple Product→Scalar Triple Product — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A plane meets the coordinate axes at A, B and C, respectively, such that the centriod of triangle ABC is (1, -2, 3). Find the equation of the plane.
→
Find the equation of a passing through the point (-1, -1, 2) and perpendicular to the planes 3x + 2y - 3z = 1 and 5x - 4y + z = 5.
→
Find the angle between the following pairs of lines:
  1. $\vec{\text{r}}=2\hat{\text{i}}-5\hat{\text{j}}+\hat{\text{k}}+\lambda\Big(3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}}\Big)\ \text{and}$
$\vec{\text{r}}=7\hat{\text{i}}-6\hat{\text{k}}+\mu\Big(\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}\Big)$
→
By using the properties of definite integrals, evaluate the integral $\int\limits_0^{\frac{\pi }{2}} {\frac{{{{\cos }^5}xdx}}{{{{\sin }^5}x + {{\cos }^5}x}}} $
→
Find the points on the line $\frac{\text{x}+2}{3}=\frac{\text{y}+1}{2}=\frac{\text{z}-3}{2}$ at a distance of 5 units from the point P(1, 3, 3).
→
Let f be an invertible real function. Write $\mathrm{(f^{-1}of)(1) + (f^{-1}of)(2) + ..... + (f^{-1}of)(100).}$
→
Evaluate the following integrals:
$\int^\limits1_{-1}5\text{x}^4\sqrt{\text{x}^5+1}\text{ dx}$
→
If D, E, F are the mid-points of side BC, CA and AB respectively of a triangle ABC, write the value of $\overrightarrow{\text{AD}}+\overrightarrow{\text{BE}}+\overrightarrow{\text{CF}}$.
→
Differentiate the function with respect to x : $\cos {x^3}{\sin ^2}\left( {{x^5}} \right)$
→
Find a solution of $ \left( x ^ { 3 } + x ^ { 2 } + x + 1 \right) \frac { d y } { d x } = 2 x ^ { 2 } + x$ which satisfy the condition $y = 1$ when $x = 0$.
→