English
Hindi

Question

Download Solution PDF

If  are coplanar, then what is  equal to?

This question was previously asked in
NDA (Held On: 18 Apr 2021) Maths Previous Year paper
Attempt in App
View all NDA Papers >
  1. 114
  2. 66
  3. 0
  4. -66

Answer (Detailed Solution Below)

Option 3 : 0
Free Tests
View all Free tests >
Free
NDA 01/2025: English Subject Test
30 Qs. 120 Marks 30 Mins
Start Now

Detailed Solution

Download Solution PDF

CONCEPT:

Properties of Scalar Triple Product

  • [a b c] = [b c a] = [c a b]
  • [a b c] = - [b a c] = - [c b a] = - [a c b]
  • [(a + b) c d] = [a c d] + [b c d]
  • [λa, b c] = λ [a b c]
  • Three non-zero vectors  are coplanar if and only if [a b c] = 0

CALCULATION:

Given:  are coplanar i.e 

⇒  and 

⇒ 

As we know that, [a b c] = [b c a] = [c a b]

⇒ 

As we know that, [λa, b c] = λ [a b c]

⇒ 

As we know that, vectors  are coplanar if and only if [a b c] = 0

⇒ 

⇒ 

Hence, correct option is  3.

Download Solution PDF
Latest NDA Updates

Last updated on Jun 18, 2025

->UPSC has extended the UPSC NDA 2 Registration Date till 20th June 2025.

-> A total of 406 vacancies have been announced for NDA 2 Exam 2025.

->The NDA exam date 2025 has been announced. The written examination will be held on 14th September 2025.

-> The selection process for the NDA exam includes a Written Exam and SSB Interview.

-> Candidates who get successful selection under UPSC NDA will get a salary range between Rs. 15,600 to Rs. 39,100. 

-> Candidates must go through the NDA previous year question paper. Attempting the NDA mock test is also essential. 

Exams Tests Super Practice

More Scalar Triple Product Questions

Q1.Let \(\rm\vec{a},\vec{b},\vec{c}\) be three non zero vectors such that \(\rm \vec{c}\) is a unit vector perpendicular to both \(\rm \vec{a}\) and \(\rm \vec{b}\). If the angle between \(\rm\vec{a}\) and \(\rm\vec{b}\) be \(\frac{\pi}{6}\) then \(\begin{bmatrix} \vec{a} &\vec{b}& \vec{c}\end{bmatrix}^2 \) is:
Q2.If \(\rm\vec{a},\vec{b}\) and \(\rm\vec{c}\) are three non-coplanar vectors, then the value of \(\rm \frac{\vec{a}.(\vec{b}\times\vec{c})}{\vec{b}.(\vec{c}\times\vec{a})}+\rm \frac{\vec{b}.(\vec{c}\times\vec{a})}{\vec{c}.(\vec{a}\times\vec{b})}+\rm \frac{\vec{c}.(\vec{b}\times\vec{a})}{\vec{a}.(\vec{b}\times\vec{c})}\) is:
Q3.The vectors λî + ĵ + 2k̂, î + λĵ − k̂ and 2î − ĵ + λk̂ are coplanar if
Q4.\(\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}\) are non-zero non-coplanar vectors and \(\overrightarrow{p}=\frac{\overrightarrow{b}\times\overrightarrow{c}}{[bca]},\overrightarrow{q}=\frac{\overrightarrow{c}\times\overrightarrow{a}}{[cab]},\overrightarrow{r}=\frac{\overrightarrow{a}\times\overrightarrow{b}}{[abc]}\), then [abc] =
Q5.Suppose \(\rm \mathop v\limits^ \to = 2\hat i + \hat j - \hat k\)  and \(\rm \mathop w\limits^ \to = \hat i + 3\hat k\). If \(\rm \mathop u\limits^ \to \) is unit vector, then the maximum value of scalar triple product \(\rm \left[ {\mathop u\limits^ \to \mathop v\limits^ \to \mathop w\limits^ \to } \right]\) is -
Q6.Let \(\rm\vec {a}, \vec{b}\) and \( \rm \vec {c}\) be three vectors such that  \(\rm\vec {a}, \vec{b}\) and \( \rm \vec {c}\)  are co-planar. Which of the following is/are correct? 1. \(\rm(\vec{a}\times \vec{b})\times \vec{c}\) is co-planar with \(\rm\vec {a}\) and \(\rm\vec {b}\) 2. \(\rm(\vec{a}\times \vec{b})\times \vec{c}\) is perpendicular to \(\rm\vec {a}\) and \(\rm\vec {b}\) Select the correct answer using the code given below.
Q7.Find the value of 'a' such that the vectors 2î - ĵ + k̂, î + 2ĵ - 3k̂ and 3î + aĵ + 5k̂ are coplanar.
Q8.Find the scalar triple product of the vectors \(\vec a = 2\hat i + 3\hat j + 4\hat k\;,\;\vec b = 3\hat i + 4\hat j + 6\hat k\;and\;\vec c = 2\hat i + 4\hat j + \hat k\)
Q9.Consider the following statements in respect of a vector \(\vec c=\vec a+\vec b\), where \(|\vec a|=|\vec b|\ne0\): 1. \(\vec c\) is perpendicular to \((\vec a-\vec b).\) 2. \(\vec c\) is perpendicular to \(\vec a \times \vec b.\) Which of the above statement is/are correct?

More Vector Algebra Questions

Q1.The vectors \(\rm \vec a, \vec b\ and\ \vec c\) are of the same length. If taken pairwise they form equal angles. If \(\rm \vec a=̂ i+̂ j\ and \ \vec b=̂ j+̂ k,\) then what can \(\vec c\) be equal to?  I. î + k̂  II. \(\rm \frac{-\hat i+4\hat j-\hat k}{3}\) Select the correct answer using the code given below. 
Q2.Let ABCDEF be a regular hexagon. If \(\rm \vec{AD}=m \vec {BC}\ and\ \vec {CF}=n\vec {AB}\) then what is mn equal to 
Q3.Let θ be the angle between two unit vectors \(\rm \vec a\ and\ \vec b. \ if\ \vec a+2\vec b\) is perpendicular to \(\rm 5\vec a-4\vec b\) then what is cos θ + cos 2θ equal to?
Q4.For what value of the angle between the vectors \(\rm \vec a\ and \ \vec b\) is the quantity \(\rm |\vec a\times \vec b|+\sqrt3|\vec a.\vec b|\) maximum? 
Q5.What is 3α + 2β equal to if (2î + 6ĵ + 27k̂) × (î + αĵ + βk̂) is a null vector?
Q6.A vector \(\rm \vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}(\alpha, \beta \in {R}) \) lies in the plane of the vectors, \(\rm \vec{b}=\hat{i}+\hat{j}\) and \(\rm \vec{c}=\hat{i}-\hat{j}+4 \hat{k}\). If \(\rm \vec{a}\) bisects the angle between \(\rm \vec{b}\) and \(\rm \vec{c}\), then
Q7.The vector equation of the plane passing through the intersection of the planes \(\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=1\) and \(\vec{r} \cdot(\hat{i}-2 \hat{j})=-2\) and the point (1, 0, 2) is:
Q8.Let ABC be a triangle such that \(\overrightarrow{B C}=\vec{a}, \overrightarrow{C A}=\vec{b}, \overrightarrow{A B}=\vec{c},|\vec{a}|=6 \sqrt{2},|\vec{b}|=2 \sqrt{3}\) and \(\vec{b} \cdot \vec{c}=12\) Consider the statements : \(\text { (S1): }|(\vec{a} \times \vec{b})+(\vec{c} \times \vec{b})|-|\vec{c}|=6(2 \sqrt{2}-1)\) \(\text { (S2): } \angle A C B=\cos ^{-1}\left(\sqrt{\frac{2}{3}}\right)\) Then
Q9.Let a unit vector which makes an angle of 60° with 2î + 2ĵ - k̂ and an angle of 45° with î - k̂ be \(\vec{C}\). Then \(\vec{C}+ \left(-\frac{1}{2}\hat{i} + \frac{1}{3\sqrt{2}}\hat{j} - \frac{\sqrt{2}}{3}\hat{k}\right)\) is:
Q10.If a̅, b̅, c̅, d̅ are the position vectors of the points A, B, C, D, respectively, such that no three of them are collinear and a̅ + c̅ = b̅ + d̅. then the quadrilateral ABCD is: 
Suggested Test Series
View All >
Current Affairs (CA) 2025 Mega Pack for SSC/Railways/State Exam Mock Test
431 Total Tests with 4 Free Tests
Start Free Test
General Knowledge for All Defence Exams
317 Total Tests with   4 Free Tests
Start Free Test

NDA Important Links

ResultAdmit CardApplication FormEligibilitySalarySyllabus

More Mathematics Questions

Q1.If cosec θ + cot θ = p, then \(\rm\frac{p^2-1}{p^2+1}\) equal:
Q2.In a binomial distribution, if the mean is 6 and the standard deviation is √2, then what are the values of the parameters n and p respectively?
Q3.Three perfect dice D1, D2 and D3 and are rolled. Let x, represent the numbers on D1, D2 and D3 respectively. What is the number of possible outcomes such that x < y < z?
Q4.Three perfect dice are rolled. Under the condition that no two show the same face, what is the probability that one of the faces shown is an ace (one)?
Q5.The occurrence of a disease in an industry is such that the workers have 20% chance of suffering from it. What is the probability that out of 6 workers chosen at random, 4 or more will suffer from the disease?
Q6.Two perfect dice are thrown. What C is the probability that the sum of the numbers on the faces is neither 9 nor 10?
Q7.Urn A contains 2 white and 2 black balls while urn B contains 3 white and 2 black balls. One ball is transferred from urn A to urn B and then a ball is drawn out of urn B. What is the probability that the ball is white?
Q8.A fair coin is tossed till two heads occur in succession. What is the probability that the number of tosses required is less than 6?  
Q9.The mean of the series x1, x2...xn is x̅ If xn is replaced by k, then what is the new mean?
Q10.Let \(\rm \Sigma_{i=1}^9x_i^2=885\) If M is the mean and σ is the standard deviation of  x1, x2, x3.....x9 then what is the value of M2 + σ2?
Testbook Edu Solutions Pvt. Ltd.
1st & 2nd Floor, Zion Building,
Plot No. 273, Sector 10, Kharghar,
Navi Mumbai - 410210
support@testbook.com
Toll Free:1800 833 0800
Office Hours: 10 AM to 7 PM (all 7 days)
Company
About usCareers We are hiringTeach Online on TestbookPartnersMediaSitemap
Products
Test SeriesTestbook PassOnline CoursesOnline VideosPracticeBlogRefer & EarnBooks
Copyright © 2014-2022 Testbook Edu Solutions Pvt. Ltd.: All rights reserved
User PolicyTermsPrivacy
Hot Links： teen patti 51 bonus teen patti rummy 51 bonus teen patti yas teen patti master real teen patti