Download App

Vector or Cross Product — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSVector or Cross Product5 Marks
Question
Find a unit vector perpendicular to the plane containing the vectors $\vec{\text{a}}=2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}.}$
✓

Answer

A vector perpendicular to the plane containing the vector $\vec{\text{a}}$ and $\vec{\text{b}}$ is given by $\vec{\text{a}}\times\vec{\text{b}}=\pm\vec{\text{c}}$ (say)
$\vec{\text{a}}\times\vec{\text{b}}=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\\text{a}_1&\text{b}_1&\text{c}_1\\\text{a}_2&\text{b}_2&\text{c}_2 \end{vmatrix}$
$\vec{\text{c}}=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\2&1&1\\1&2&1 \end{vmatrix}$
$\vec{\text{c}}=\hat{\text{i}}(1-2)-\hat{\text{j}}(2-1)+\hat{\text{k}}(4-1)$
$\vec{\text{c}}=-\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$
$\hat{\text{c}}=\frac{\vec{\text{c}}}{|\vec{\text{c}}|}$
$=\frac{-\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}}{\sqrt{(-1)^2+(-1)^2+(3)^2}}$
$=\frac{-\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}}{\sqrt{1+1+9}}$
$=\frac{1}{\sqrt{11}}\big(-\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}\big)$
Unit vector perpendicular to the plane of $\vec{\text{a}}$ and $\vec{\text{b}}=\pm\frac{1}{\sqrt{11}}\big(-\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}\big).$

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 "5 Marks Questions" from Vector or Cross Product→Vector or Cross Product — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

If $x^y + y^x = (x + y)^{x+y},$ find $\frac{\text{dy}}{\text{dx}}$
→
Evaluate: $\int\limits_{0}^{\pi}\frac{4\text{x}\sin\text{x}}{1 + \cos^{2}\text{x}}\text{dx}.$
→
Find the area of the ragion in the first quadrant bounded by the parabola $y = 4x^2$ and the lines $x = 0, y = 1$ and $y = 4.$
→
Explain why the experiment of tossing a coin three times is said to have binomial distribution.
→
Find the vector equation of the plane passing through three point with position vectors $\hat{\text{i}}+\hat{\text{j}}-2\hat{\text{k}},2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and $\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}.$ Also, find coordinates of the point of intersection of this plane and the line $\vec{\text{r}}=3\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}+\lambda(2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}).$
→
Differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{y}=0,\text{y}(0)=1,\text{y}'(0)=1$Function $\text{y}=\sin\text{x}+\cos\text{x}$
→
Find the foot of the perpendicular from (0, 2, 7) on the line $\frac{\text{x}+2}{-1}=\frac{\text{y}-1}{3}=\frac{\text{z}-3}{-2}.$
→
Compute the elements $a_{43}$ and $a_{22}$ of the matrix:
$\text{A}=\begin{bmatrix}0&1&0\\2&0&2\\0&3&2\\4&0&4\end{bmatrix}\begin{bmatrix}2&-1\\-3&2\\4&3\end{bmatrix}\begin{bmatrix}0&1&-1&2&-2\\3&-3&4&-4&0\end{bmatrix}$
→
In the following, determine the values of constants involved in the definition so that the given function is continuous:
$\text{f(x)}=\begin{cases}2,&\text{if }\text{ x}\leq3\\\text{ax}+\text{b},&\text{if }3<\text{ x}<5\\9,&\text{if }\text{ x}\geq5\end{cases}$
→
Show that the points $2\hat{\text{i}},-\hat{\text{i}}-4\hat{\text{j}}\text{ and }-\hat{\text{i}}+4\hat{\text{j}}$ form an isosceles triangle.
→