Let a = i + j + k , b = i and c =c_1 i +c_2 j +c_3 k . Then, If C… - Vidyadip

Question

Let $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}$ and $\vec{\text{c}}=\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}}.$ Then,
If $C_2 = -1$ and $C_3 = 1$, show that no value of $C_1$ can make $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ coplanar.

✓

Answer

If $C_2= -1$ and $C_3 = 1$, then $\vec{\text{a}}=\hat{\text{i}}+\vec{\text{j}}+\vec{\text{k}},\vec{\text{b}}=\hat{\text{i}}$ and $\vec{\text{c}}={\text{c}}_1\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}.$
We know that vectors $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are coplanar if $\big[\vec{\text{a}}\vec{\text{b}}{\text{c}}\big]=0.$
for $\big[\vec{\text{a}},\vec{\text{b}},\text{c}\big]$ to be coplanar:
$\therefore\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]=0$
$\Rightarrow\begin{vmatrix}1&1&1\\1&0&0\\\text{c}_1&-1&1 \end{vmatrix}=0$
$\Rightarrow 1(0-0)-1(1-0)+1(-1-0)=0$
$\Rightarrow-1-1=0$
$\Rightarrow -2=0$
But this is never possible, whatever be the value of $C_1$. Thus, no value of $C_1$ can make $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ coplanar.

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 VECTOR ALGEBRA→VECTOR ALGEBRA — 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 differential equations:
$\frac{\text{dy}}{\text{dx}}=(1+\text{x}^2)(1+\text{y}^2)$

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$f(x) = x^2 + x - 1$ on $[0, 4]$

Prove that: $\int_0^\pi \frac{x}{1-\cos \alpha \sin x} d x=\frac{\pi(\pi-\alpha)}{\sin \alpha}$

Let $A = [-1, 1].$ Then, discuss whether the following functions from A to itself are one-one, onto or bijective:$h(x) = x^2$

Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,
$\text{f}(\text{x})=\text{x}\sqrt{1-\text{x}},\text{x}\leq1$

Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.

When the curve crosses the y-axis, the point on the curve is of the form $(0, y).$

If D is the mid-point of side BC of a triangle ABC such that $\overrightarrow{\text{AB}}+\overrightarrow{\text{AC}}=\lambda\overrightarrow{\text{AD}}$, write the value of $\lambda$.

Integrate the function $\sqrt{x^{2}+3 x}$

Evaluate the following integrals:
$\int\frac{\text{cosec }\text{x}}{\text{cosec }\text{x}-\cot\text{x}}\text{dx}$