Download App

Vector or Cross Product — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsVector or Cross Product5 Marks
Question
Given $\vec{\text{a}}=\frac{1}{7}\big(2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}\big),\vec{\text{b}}=\frac{1}{7}\big(3\hat{\text{i}}-6\hat{\text{j}}+2\hat{\text{k}}\big),$$\vec{\text{c}}=\frac{1}{7}\big(6\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\big),\hat{\text{i}},\hat{\text{j}},\hat{\text{k}}$
being a right handed orthogonal system of unit vector in spece, show that $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ is also another system.
✓

Answer

Given:
$\vec{\text{a}}=\frac{1}{7}\big(2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}\big)$
$\vec{\text{b}}=\frac{1}{7}\big(3\hat{\text{i}}-6\hat{\text{j}}+2\hat{\text{k}}\big)$
$\vec{\text{c}}=\frac{1}{7}\big(6\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\big)$
$\vec{\text{a}}\times\vec{\text{b}}=\Big(\frac{1}{7}\Big)\Big(\frac{1}{7}\Big)\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\2&3&6\\3&-6&2 \end{vmatrix}$
$=\frac{1}{49}\big(42\hat{\text{i}}+14\hat{\text{j}}-21\hat{\text{k}}\big)$
$=\frac{1}{49}\big[7\big(6\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\big)\big]$
$=\frac{1}{7}\big(6\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\big)$
$=\vec{\text{c}}$
$\vec{\text{b}}\times\vec{\text{c}}=\Big(\frac{1}{7}\Big)\Big(\frac{1}{7}\Big)\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\3&-6&2\\6&2&-3 \end{vmatrix}$
$=\frac{1}{49}\big(14\hat{\text{i}}+21\hat{\text{j}}+42\hat{\text{k}}\big)$
$=\frac{1}{49}\big[7\big(2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}\big)\big]$
$=\frac{1}{7}\big(2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}\big)$
$=\vec{\text{a}}$
$\vec{\text{c}}\times\vec{\text{a}}=\Big(\frac{1}{7}\Big)\Big(\frac{1}{7}\Big)\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\6&2&-3\\2&3&6 \end{vmatrix}$
$=\frac{1}{49}\big(21\hat{\text{i}}-42\hat{\text{j}}+14\hat{\text{k}}\big)$
$=\frac{1}{49}\big[7\big(3\hat{\text{i}}-6\hat{\text{j}}+2\hat{\text{k}}\big)\big]$
$=\frac{1}{7}\big(3\hat{\text{i}}-6\hat{\text{j}}+2\hat{\text{k}}\big)$
$=\vec{\text{b}}$
$|\vec{\text{a}}|=\frac{1}{7}\sqrt{4+9+36}$
$=\frac{7}{7}$
$=1$
$\big|\vec{\text{b}}\big|=\frac{1}{7}\sqrt{9+36+4}$
$=\frac{7}{7}$
$=1$
$|\vec{\text{c}}|=\frac{1}{7}\sqrt{36+4+9}$
$=\frac{7}{7}$
$=1$
Thus, $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ from a right handed orthogonal system of unit vectors.

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 Vector or Cross Product→Vector or Cross Product — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Solve the following systems of linear equations by cramer's rule:
$x + y + z + 1 = 0,$
$ax + by + cz + d = 0,$
$a^2x + b^2y + x^2z + d^2 = 0$
→
Solve the following differential equation:$\frac{\text{dy}}{\text{dx}}+\text{y}=\text{e}^{-2\text{x}}$
→
Evaluate the following intregals:
$\int\frac{1}{\text{x}(\text{x}^3+8)}\ \text{dx}$
→
Evalute the following integrals:
$\int\frac{\sin(\text{x}-\text{a})}{\sin(\text{x}-\text{b})}\text{dx}$
→
If lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect each other then find $\mathrm{k}$.
→
Evaluate the following integrals:$\int\frac{1}{\sqrt{7-3\text{x}-2\text{x}^2}}\text{ dx}$
→
If $e ^y= y ^{ x }$, then show that $\frac{d y}{d x}=\frac{(\log y)^2}{\log y-1}$
→
Find the vector and cartesian equations of the line passing through $(1, 2, 3)$ and parallel to the planes $\vec{\text{r}}\cdot(\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}})=5$ and $\vec{\text{r}}\cdot(3\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}})=6$
→
A company produces mixers and food processors. Profit on selling one mixer and one food processor is ₹ 2,000/- and ₹ 3,000/- respectively. Both the products are processed through three Machines A, B, C. The time required in hours by each product and total time available in hours per week on each machine are as follows :

Image

How many mixers and food processors should be produced to maximize the profit?

→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+2\text{y}=\text{xe}^{4\text{x}}$
→