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Given,

$\overrightarrow{a}.\overrightarrow{b}=0$ and $\overrightarrow{a}\times \overrightarrow{b}=0$

For,

$\overrightarrow{a}.\overrightarrow{b}=0$ , then either $\left|\overrightarrow{a}\right|=0$ or $\left|\overrightarrow{b}\right|=0$ or $\overrightarrow{a}\perp \overrightarrow{b}$

For,

$\overrightarrow{a}\times \overrightarrow{b}=0$ , then either $\left|\overrightarrow{a}\right|=0$ or $\left|\overrightarrow{b}\right|=0$ or $\overrightarrow{a}?\overrightarrow{b}$

$\therefore$ In case $\overrightarrow{a}$ and $\overrightarrow{b}$ are non- zero on both side.

But $\overrightarrow{a}$ and $\overrightarrow{b}$ cannot be both perpendicular and parallel simultaneously.

So, we can conclude that

$\left|\overrightarrow{a}\right|=0$ or $\left|\overrightarrow{b}\right|=0$

Yes, there are many government Optometry colleges in India. Some of them are mentioned below along with their tuition fees:

Disclaimer: This information is sourced from official website and may vary.

Here are some top Sound Editing in Music & Sound Production colleges in India, offering full-time courses:

Note: The information is sourced from the official/counselling body of the institute/exam.

The college offers various courses such as BArch, MArch, BSc, etc. The selection criteria are a mix of entrance exams and merit-based, as per respective courses. To get admitted to SCAD, candidates must meet the eligibility criteria set by the college. Apart from the UG and PG courses, the college also offers a Diploma programme. The application process at Saveetha College of Architecture and Design is conducted online. The courses are offered in various specialisations such as Fashion Design, Urban, Industrial / Product Design, etc.

$\left(2\widehat{i}+6\widehat{j}+27\widehat{k}\right)\times \left(\widehat{i}+\lambda \widehat{j}+\mu \widehat{k}\right)=\overrightarrow{0}$

$\begin{array}{l}\Rightarrow \widehat{i}\left(6\mu -27\lambda \right)-\widehat{j}\left(2\mu -27\right)+\widehat{k}\left(2\lambda -6\right)\\ =0\widehat{i}+0\widehat{j}+0\widehat{k}\end{array}$

On comparing both side components,

$\begin{array}{l}6\mu -27\lambda =0,\\ 2\mu -27=0\\ 2\mu =27\\ \mu =\frac{27}{2},\\ 2\lambda -6=0\\ 2\lambda =6\\ \lambda =\frac{6}{2}=3\end{array}$

$\therefore$ Therefore, the value of $\mu =\frac{27}{2}$ and $\lambda =3$

Yes, there are many private Optometry colleges in India. Some of them are mentioned below along with their tuition fees:

Disclaimer: This information is sourced from official website and may vary.

Show that

$\begin{array}{l}\left(\overrightarrow{a}-\overrightarrow{b}\right)\times \left(\overrightarrow{a}+\overrightarrow{b}\right)=2\left(\overrightarrow{a}\times \overrightarrow{b}\right)\\ \left(\overrightarrow{a}-\overrightarrow{b}\right)\times \left(\overrightarrow{a}+\overrightarrow{b}\right)\\ =\overrightarrow{a}\left(\overrightarrow{a}+\overrightarrow{b}\right)-\overrightarrow{b}\left(\overrightarrow{a}+\overrightarrow{b}\right)\\ =\overrightarrow{a}\times \overrightarrow{a}+\overrightarrow{a}\times \overrightarrow{b}-\overrightarrow{b}\times \overrightarrow{a}-\overrightarrow{b}\times \overrightarrow{b}\\ =0+\overrightarrow{a}\times \overrightarrow{b}-\overrightarrow{b}\times \overrightarrow{a}-0\\ =\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{b}\left[\overrightarrow{a}\times \overrightarrow{b}=-\overrightarrow{b}\times \overrightarrow{a}\right]\\ =2(\overrightarrow{a}\times \overrightarrow{b})\end{array}$

The DSSSB exam was conducted from July 7 to August 4, 2025 in multiple shifts for PGT posts. Candidates can check below the methods to download the exam schedule.
- Visit the official website dsssb.delhi.gov.in
- Check the latest notification regarding the exam dates
- Click on the notification
- The exam schedule will be available in PDF format containing the post-wise exam dates
- Search for your applied posts and check exam date, shift-timings and subject details
- Keep a hard copy for future reference

Saveetha College of Architecture and Design admissions 2025 are currently ongoing. Interested candidates can apply in online mode through the official website (simatsadmissions.saveetha.com/). The online application process is explained in the following steps:

Step 1: Visit the official website of Saveetha College of Architecture and Design.

Step 2: Fill out the form.

Step 3: Complete the form and submit it.

IIT Madras is an renowned institute worldwide which offers BTech courses based on JEE Advanced exam scores. It provides admission for various BTech and Integrated courses including CSE, ECE, and Civil Engigineering programs. In 2024, IIT Madras has released the JEE Advanced cutoff 2024. In connection to it, the CSE last-round cutoff rank stood at 159 for the General AI category candidates.

Candidates can check out the five-year CSE cutoff trend from the table below for the General AI category:

While analysing the data, a fluctuating trend over the period of five years can be visible in the IIT Madras cutoffs.

Let $\overrightarrow{a}=\left(a_1,a_2,a_3\right)$ as component

We know,

$\overrightarrow{a}$ is a unit vector, $\left|\overrightarrow{a}\right|=1$

Given that,

$\overrightarrow{a}$ marks angles $\frac{\pi }{3}$ with $\widehat{i}$ , $\frac{\pi }{4}$ with $\widehat{j}$ and $\theta$ with $\widehat{k}$ acute angle.

Now,

$cos\frac{\pi }{3}=\frac{a_1}{\left|\overrightarrow{a}\right|}\Rightarrow \frac{1}{2}=a_1\left[\left|\overrightarrow{a}\right|=1\right]\\ cos\frac{\pi }{4}=\frac{a_2}{\left|\overrightarrow{a}\right|}\Rightarrow 1/\surd 2$
$\begin{array}{l}\frac{\mathrm{}}{}=a_2\\ cos\theta =\frac{a_3}{\left|\overrightarrow{a}\right|}\Rightarrow a_3=cos\theta \end{array}$

We know,

$\left|\overrightarrow{a}\right|=1$

Yes, applications are open at Saveetha College of Architecture and Design for various courses. The mode of application is online. Aspirants need to apply at the official website, i.e., simatsadmissions.saveetha.com/ for admission to the various programme. Candidates can fill out the application form on the institute's official website.

Given,

$\begin{array}{l}\overrightarrow{a}=3\widehat{i}+2\widehat{j}+2\widehat{k}\\ \overrightarrow{b}=\widehat{i}+2\widehat{j}-2\widehat{k}\\ \overrightarrow{a}+\overrightarrow{b}=4\widehat{i}+4\widehat{j},\overrightarrow{a}-\overrightarrow{b}=2\widehat{i}+4\widehat{j}\end{array}$

A vector which is perpendicular to both $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-\overrightarrow{b}$ is given by

Say

Therefore, the unit vector is

$\begin{array}{l}\frac{\overrightarrow{c}}{\left|\overrightarrow{c}\right|}=\pm \frac{16\widehat{i}-16\widehat{j}-8\widehat{k}}{24}\\ =\pm \frac{16}{24}\widehat{i}\pm \frac{16}{24}\widehat{j}\pm \frac{8}{24}\widehat{k}\\ =\pm \frac{2}{3}\widehat{i}\pm \frac{2}{3}\widehat{j}\pm \frac{1}{3}\stackrel{}{k}\end{array}$

In order to get admission at Saveetha College of Architecture and Design candidates must keep certain documents ready. Students can check the list of documents below:

• Scanned Class 10, Class 12 and graduation marksheet
• Leaving Certificate
• Aadhaar Card
• Passport-size photograph
• Signature

Note: The list of documents mentioned above are the general documents. For more details visit the official website.

As per the the latest student enrollment data (2023), made available by GMU, the number of students currently pursuing their higher studies as part of the university campus stands at 40,200, which is a record for any university located in the Virginia region.

These include: 27,100 students enrolled as part of undergraduate courses, while about 7,700 students are enrolled in postgraduate studies at GMU.

The institute offers one of the largest course catalogues in the region, comprising around 200 UG & PG degree courses, & 100+ certification courses at various levels.

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Both Sound Editing and Sound Engineering are different in terms of their role and focus. Sound Editing is about audio manipulation, clearing & enhancing audio quality, shaping sounds, etc. Meanwhile, Sound Engineering means editing, mixing, reproducing, and delivering the final track. It covers the technical aspects of sound production.