Scalar Triple Product
Class 12th Mathematics RD Sharma Volume 2 Solution
- [i j k]+[j k i]+[k i j] Evaluate the following :
- [2 i j k]+[i k j]+[k j2 i] Evaluate the following :
- Find [lll vector a & vector b & vector c] when vector a = 2 i-3 j , vector b =…
- Find [lll vector a & vector b & vector c] when vector a = i-2 j+3 k , vector b…
- vector a = 2 i+3 j+4 k , vector b = i+2 j - k , vector c = 3 i - j+2 k Find…
- vector a = 2 i-3 j+4 k , vector b = i+2 j - k vector c = 3 i - j-2 k Find the…
- vector a = 11 i , vector b = 2 j , vector c = 13 k Find the volume of the…
- vector a = i + j + k , vector b = i - j + k , vector c = i+2 j - k Find the…
- vector a = i+2 j - k , vector b = 3 i+2 j+7 k , vector c = 5 i+6 j+5 k Show…
- vector a = - 4 i-6 j-2 k , vector b = - i+4 j+3 k , vector c = - 8 i - j+3 k…
- a = i-2 j+3 k , b = - 2 i+3 j-4 k , c = i-3 j+5 k Show that each of the…
- vector a = i - j + k , vector b = 2 i + j - k , vector c = lambda i - j +…
- vector a = 2 i - j + k , vector b = i+2 j-3 k , vector c = lambda i + lambda…
- vector a = i+2 j-3 k , vector b = 3 i + lambda j + k , vector c = i+2 j+2 k…
- vector a = i+3 j , vector b = 5 k , vector c = lambda i - j Find the value of…
- Show that the four points having position vectors 6 i-7 j , 16 i-19 j-4 k , 3…
- Show that the points A(- 1, 4, - 3), B(3, 2, - 5), C(- 3, 8, - 5) and D(- 3, 2,…
- Show that four points whose position vectors are 6 i-7 j , 16 i-19 j-4 k , 3…
- Find the value of for which the four points with position vectors - j - k , 4…
- Prove that : - (vector a - vector b) (vector b - vector c) x (vector c -…
- vector a , vector b and vector c are the position vectors of points A, B and C…
- Let vector a = 1 + j + k , vector b = 1 and c = c_1 i+c_2 j+c_3 k Then, If…
- Let vector a = 1 + j + k , vector b = 1 and c = c_1 i+c_2 j+c_3 k Then, If…
- Find for which the points A(3, 2, 1), B(4, λ, 5), C(4, 2, - 2) and D(6, 5, -…
- If four points A, B, C and D with position vectors 4 i+3 j+3 k , 5 1+x j+7 k 5…
- Write the value of [ {lll} { 2 {i} } & { 3 hat{j} } & { 4 hat{k} } ]…
- Write the value of [ {i} + hat{j} hat{j} + hat{k} hat{k} - hat{i}]…
- Write the value of [ {i} - hat{j} hat{j} - hat{k} hat{k} - hat{i}]…
- Find the values of ‘a’ for which the vectors vector { alpha } = {i}+2 hat{j} +…
- Find the volume of the parallelepiped with its edges represented by the vectors {i} +…
- If vector {a} , vec{b} are non-collinear vectors, then find the value of [ vector {a}…
- If the vectors (sec2 A) {i} + hat{j} + hat{k} , hat{i} + (sec^{2}b) hat{j} + hat{k} ,…
- For any two vectors of vector {a} vec{b} of magnitudes 3 and 4 respectively, write the…
- If [3 vector {a}7 vec{b} vec{c} vec{d}] = lambda [ vec{a} vec{b} vec{c}] + μ [ vec{b}…
- If vector {a} , vec{b} , vec{c} are non-coplanar vectors, then find the value of…
- Find vector {a} c. ( vec{b} x vec{c} ) , if vector {a} = 2 {i} + hat{j}+3 hat{k}…
- If vector {a} lies in the plane of vectors vector {b} and vector {c} , then which…
- The value of [ vector {a} - vec{b} vec{b} - vec{c} vec{c} - vec{a}] , where | vector…
- If vector {a} , vector {b} , vector {c} are three non-coplanar mutually…
- If vector {r} c. vec{a} = vec{r} vec{b} = vec{r} cdot vec{c} = 0 for some non-zero…
- For any three vector vector {a} , vec{b} , vec{c} the expression ( vector {a} -…
- If vector {a} , vector {b} , vector {c} are non-coplanar vectors, then is Mark the…
- Let vector {a} = a_{1} {i}+a_{2} hat{j}+a_{3} hat{k} , vec{b} = b_{1} hat{i}+b_{2}…
- If vector {a} = 2 {i}-3 hat{j}+5 hat{k} , vec{b} = 3 hat{i}-4 hat{j}+5 vec{k} and…
- If [2 vector {a}+4 vec{b} vec{c} vec{d}] = lambda [ vec{a} vec{c} vec{d}] + μ [ vec{b}…
- [ vector {a} vec{b} vec{a}x vec{b}] + ( vec{a} c. vec{b} ) ^{2} Mark the correct…
- If the vectors 4 {i}+11 hat{j}+m hat{k} , 7 hat{i}+2 hat{j}+6 hat{k} and {i}+5…
- For non-zero vectors a⃗, b⃗ and c⃗ the relation | ( vector {a} x vec{b} ) c.…
- ( vector {a} + vec{b} ) c. ( vec{b} + vec{c} ) x ( vec{a} + vec{b} + vec{c} ) = Mark…
- If a⃗ ,b⃗,c⃗ are three non-coplanar vectors, then ( vector {a} + vec{b} + vec{c} )…
- ( vector {a}+2 vec{b} - vec{c} ) c. { ( vec{a} - vec{b} ) x ( vec{a} - vec{b} -…
Exercise 26.1
Question 1.Evaluate the following :
Answer:
Formula: –
(i)
(ii)
(iii) 
we have
using Formula(i) and (iii)
therefore, using Formula (ii)
Question 2.
Evaluate the following :
Answer:
Formula: -
(i) 
(ii) 
(iii)
Given: -
we have
using Formula (i)
using Formula (ii)
therefore,
Question 3.
Find 
when
and 
Answer:
Formula: -
(i) 
(ii) 
Given: -
using Formula(i)
now, using
= 2( – 1 – 0) + 3( – 1 + 3)
= – 2 + 6
= 4
therefore,
Question 4.
Find when
and 
Answer:
Formula: -
(i) If 
and
then,
(ii)
Given: -
now, using
= 1(1 + 1) + 2(2 + 0) + 3(2 – 0)
= 2 + 4 + 6 = 12
therefore,
Question 5.
Find the volume of the parallelepiped whose coterminous edges are represented by the vectors:
Answer:
Formula : -
(i) if
(ii)
Given: -
we know that the volume of parallelepiped whose three adjacent edges are
we have
now, using
= 2(4 – 1) – 3(2 + 3) + 4( – 1 – 6)
= – 37
therefore, the volume of the parallelepiped is
Question 6.
Find the volume of the parallelepiped whose coterminous edges are represented by the vectors:
Answer:
Formula : -
(i) ifand
then,
(ii)
Given: -
we know that the volume of parallelepiped whose three adjacent edges are
we have
now, using
= 2( – 4 – 1) – 3( – 2 + 3) + 4( – 1 – 6)
= – 35
therefore, the volume of the parallelepiped is
Question 7.
Find the volume of the parallelepiped whose coterminous edges are represented by the vectors:
Answer:
Formula : -
Given: -
we know that the volume of parallelepiped whose three adjacent edges are
we have
now, using
= 11(26 – 0) + 0 + 0 = 286
therefore, the volume of the parallelepiped is
[a⃗ b⃗ c⃗] = |286| = 286 cubic unit.
Question 8.
Find the volume of the parallelepiped whose coterminous edges are represented by the vectors:
Answer:
Formula: -
and
then,
Given: -
we know that the volume of parallelepiped whose three adjacent edges are
is equal to 
.
we have
now, using
= 1(1 – 2) – 1( – 1 – 1) + 1(2 + 1)
= 4
therefore, the volume of the parallelepiped is
= 4 cubic unit.
Question 9.
Show that each of the following triads of vectors is coplanar :
Answer:
Formula : –
(iii)Three vectors a ⃗,b ⃗, and c ⃗are coplanar if and only if
Given: -
we know that three vector 
are coplanar if their scalar triple product is zero
we have
using
= 1(10 – 42) – 2(15 – 35) – 1(18 – 10)
= 0.
Hence, the Given vector are coplanar.
Question 10.
Show that each of the following triads of vectors is coplanar :
Answer:
Formula : -
Given: -
we know that three vector are coplanar if their scalar triple product is zero
we have
now, using
= – 4(12 + 13) + 6( – 3 + 24) – 2(1 + 32)
= 0
hence, the Given vector are coplanar.
Question 11.
Show that each of the following triads of vectors is coplanar :
Answer:
Formula : -
and then,
(iii) Three vectors 
are coplanar if and only if a⃗.(b⃗×c⃗) = 0
Given: -
we know that three vector a⃗,b⃗,c⃗ are coplanar if their scalar triple product is zero
we have
now, using
= 1(15 – 12) + 2( – 10 + 4) + 3(6 – 3)
= 3 – 12 + 9 = 0
Question 12.
Find the value of λ so that the following vectors are coplanar.
Answer:
Formula : -
Given: -
we know that vector are coplanar if their scalar triple product is zero
we have
now, using
⇒ 0 = 1(λ – 1) + 1(2λ + λ) + 1( – 2 – λ)
⇒ 0 = λ – 1 + 3 λ – 2 – λ
⇒ 0 = 3 λ – 3
⇒ λ = 1
Question 13.
Find the value of λ so that the following vectors are coplanar.
Answer:
Formula : -
Given: -
we know that vector are coplanar if their scalar triple product is zero
we have
now, using
⇒ 0 = 2(10 + 3λ) + 1(5 + 3λ) + 1(λ – 2λ)
⇒ 0 = 8 λ + 25
Question 14.
Find the value of λ so that the following vectors are coplanar.
Answer:
Formula : -
Given: -
we know that vector 
are coplanar if their scalar triple product is zero
we have
now, using
⇒ 0 = 1(2 λ – 2) – 2(6 – 1) – 3(6 – λ)
⇒ 0 = 5 λ – 30
⇒ λ = 6
Question 15.
Find the value of λ so that the following vectors are coplanar.
Answer:
Formula : -
andthen,
Given: -
we know that vector are coplanar if their scalar triple product is zero
we have
now, using
⇒ 0 = 1(0 + 5) – 3(0 – 5λ) + 0
⇒ 0 = 5 + 15λ
Question 16.
Show that the four points having position vectors 
are not coplanar.
Answer:
Formula : -
andthen,
and
then
Given: -
The four points are coplaner if vector AB⃗,AC⃗,AD⃗ are coplanar.
now, using
= 10(100 + 72) + 12( – 60 – 24) – 4( – 72 + 40) = 840
≠0.
hence the point are not coplanar
Question 17.
Show that the points A( – 1, 4, – 3), B(3, 2, – 5), C( – 3, 8, – 5) and D( – 3, 2, 1) are coplanar.
Answer:
Formula: -
Given: -
AB = position vector of B - position vector of A
AC = position vector of c - position vector of A
AD = position vector of c - position vector of A
The four pint are coplanar if the vector are coplanar.
thus,
now, using
= 4(16 - 4) + 2( - 8 - 4) - 2( - 4 + 8) = 0
hence proved.
Question 18.
Show that four points whose position vectors are 
are coplanar.
Answer:
Formula : -
and
then
(iii) Three vectors
are coplanar if and only if
(iv) If 
andthen,
let
The four points are coplanar if the vector 
are coplanar.
now, using
= 10(100 + 12) + 12( – 60 – 24) – 4( – 12 + 40) = 0.
hence the point are coplanar
Question 19.
Find the value of for which the four points with position vectors 
and 
are coplanar.
Answer:
Formula : -
and
then
(iii) Three vectors
are coplanar if and only if 
(iv) ifandthen,
Given: -
The four points are coplaner if vector AB⃗,AC⃗,AD⃗ are coplanar.
now, using
⇒ 4(50 – 25) – 6(15 + 20) + (λ + 1)(15 + 40) = 0.
⇒ λ = 1
hence the point are coplanar
Question 20.
Prove that : -
Answer:
Formula: -
taking L.H.S
using Formula (i)
using Formula(ii)
L.H.S = R.H.S
Question 21.
and 
are the position vectors of points A, B and C respectively, prove that : 
is a vector perpendicular to the plane of triangle ABC.
Answer:
if 
represents the sides AB,
if 
represent the sides BC,
if 
respresent the sidesAC of triangle ABC
is perpendicular to plane of triangle ABC. …… (i)
is perpendicular to plane of triangle ABC. …… (ii)
is perpendicular to plane of triangle ABC. …… (iii)
adding all the (i) + (ii) + (iii)
hence 
is a vector perpendicular to the plane of the triangle ABC
Question 22.
Let 
and 
Then,
If 
and 
find 
which makes 
and 
coplanar.
Answer:
Formula: -
Given: -
are coplanar if
now, using
⇒ 0 – 1(c3) + 1(2) = 0
⇒ c3 = 2
Question 23.
Let and Then,
If and 
show that no value of 
can make and coplanar.
Answer:
Formula: -
and
then,
(iii)Three vectors 
are coplanar if and only if
we know that are coplanar if
now, using
⇒ 0 – 1(c3) + 1(2) = 0
⇒ c3 = 2
Question 24.
Find for which the points A(3, 2, 1), B(4, λ, 5), C(4, 2, – 2) and D(6, 5, – 1) are coplanar.
Answer:
Formula: -
and then,
let position vector of
position vector of
position vector of
position vector of
The four points are coplanar if the vector 
are coplanar
now, using
⇒ 1(9) – (λ – 2)( – 2 + 9) + 4(3 – 0) = 0
⇒ 7λ = 35
⇒ λ = 5
Question 25.
If four points A, B, C and D with position vectors 
and 
respectively are coplanar, then find the value of x.
Answer:
Formula: -
(iii) Three vectors a⃗ ,b⃗ , and c⃗ are coplanar if and only if a⃗.(b⃗×c⃗) = 0
let position vector of
position vector of
position vector of
position vector of
The four points are coplanar if the vector 
are coplanar
⇒ 1(9) – (x – 2)( – 2 + 9) + 4(3) = 0
⇒ 9 – 7x + 14 + 12 = 0
⇒ 35 = 7x
⇒ x = 5
Very Short Answer
Question 1.Write the value of 
Answer:
The meaning of the notation 
is the scalar triple product of the three vectors; which is computed as 
So we have 
(
)
Question 2.
Write the value of 
Answer:
Here we have 
= 
Question 3.
Write the value of 
Answer:
The value of the above product is the value of the matrix 
Question 4.
Find the values of ‘a’ for which the vectors 
and 
are coplanar.
Answer:
Three vectors are coplanar iff (if and only if) 
Hence we have value of the matrix 
We have 2a2-3a+1=0
2a2-2a-a+1=0
Solving this quadratic equation we get 
Question 5.
Find the volume of the parallelepiped with its edges represented by the vectors 
Answer:
Volume of the parallelepiped with its edges represented by the vectors 
is 
=
=
Question 6.
If 
are non-collinear vectors, then find the value of 
Answer:
for any vector 
We have 
Replacing 
=
Question 7.
If the vectors (sec2 A) 
are coplanar, then find the value of cosec2A A + cosec2B + cosec2C.
Answer:
For three vectors to be coplanar we have 
Which gives 
………(1)
………(2)
Substituting equation 2 in 1 we have
Let 
So we have 
=(x-2)(y-2)(z-2)-(x-2)(y-1)(z-1)-(x-1)(y-2)(z-1)-(x-1)(y-1)(z-2)+2(x-1)(y-1)(z-1)=0
Solving we have x+y+z=4
Hence cosec2A + cosec2B + cosec2C = 4
Question 8.
For any two vectors of 
of magnitudes 3 and 4 respectively, write the value of 
Answer:
the dot and cross can be interchanged in scalar triple product.
Let the angle between 
vector be 
=144 sin2 θ+144 cos2 θ
=144(1)
=144
Question 9.
If 
then find the value of λ + μ.
Answer:
λ = 3
μ = 1
So, λ + μ = 3 + 1
= 4
Question 10.
If 
are non-coplanar vectors, then find the value of 
Answer:
the dot and cross can be interchanged in scalar triple product.
Also 
(cyclic permutation of three vectors does not change the value of the scalar triple product)
=
Using these results 
Question 11.
Find 
, if 
and 
Answer:
Mcq
Question 1.Mark the correct alternative in each of the following:
If 
lies in the plane of vectors 
and 
, then which of the following is correct?
A. 
B. 
C. 
D. 
Answer:
Here, 
lies in the plane of vectors 
and 
, which means , and are coplanar.
We know that 
is perpendicular to and .
Also dot product of two perpendicular vector is zero.
Since, , , are coplanar, is perpendicular to .
So, 
Question 2.
Mark the correct alternative in each of the following:
The value of 
, where 
,
,
is
A. 0
B. 1
C. 6
D. none of these
Answer:
= 0
Question 3.
Mark the correct alternative in each of the following:
If 
,
,
are three non-coplanar mutually perpendicular unit vectors, then 
is
A. ±1
B. 0
C. –2
D. 2
Answer:
Here, ⊥ ⊥ and 
.
⇒ angle between 
and is 
or 
.
= ±1
Question 4.
Mark the correct alternative in each of the following:
If 
for some non-zero vector 
, then the value of , is
A. 2
B. 3
C. 0
D. none of these
Answer:
Here, 
, and are coplanar.
Question 5.
Mark the correct alternative in each of the following:
For any three vector 
the expression 
equals
A.
B. 
C. 
D. none of these
Answer:
= 
= 0
Question 6.
Mark the correct alternative in each of the following:
If , 
, 
are non-coplanar vectors, then 
is
A. 0
B. 2
C. 1
D. none of these
Answer:
=0
Question 7.
Mark the correct alternative in each of the following:
Let 
and 
be three non-zero vectors such that c⃗ is a unit vector perpendicular to both a⃗ and b⃗ . If the angle between a⃗ and b⃗ is
, then 
is equal to
A. 0
B. 1
C.
D. 
Answer:
(∵ c⃗ is perpendicular to a⃗ and b⃗ ⇒ angle is 0)
(∵ c⃗ is unit vector )
Question 8.
Mark the correct alternative in each of the following:
If 
and 
then the volume of the parallelepiped with conterminous edges 
, 
, 
is
A. 2
B. 1
C. –1
D. 0
Answer:
Let 
Now,the volume of the parallelepiped with conterminous edges 
, 
,
is given by
=5× (-21+18)+7× (24-21)+10× (-48+49) ×
=5× (-3)+7× 3+10× 1
=-15+21+10
=16
Question 9.
Mark the correct alternative in each of the following:
If 
then 
A. 6
B. –6
C. 10
D. 8
Answer:
Now, comparing the coefficient of lhs and rhs we get, λ=2 and μ=4
∴ λ + μ = 2+4
=6
Question 10.
Mark the correct alternative in each of the following:
A. 
B. 
C.
D. 2
Answer:
Question 11.
Mark the correct alternative in each of the following:
If the vectors 
and 
are coplanar, then m =
A. 0
B. 38
C. –10
D. 10
Answer:
Here, vector a, b, and c are coplanar. So, 
.
∴ 
∴ 4(8-30)-11(28-6)+m(35-2)= 0
∴ 4(-22)-11(22)+33m = 0
∴ -88 -242 +33m = 0
∴ 33m = 330
∴ m = 10
Question 12.
Mark the correct alternative in each of the following:
For non-zero vectors a⃗, b⃗ and c⃗ the relation 
holds good, if
A. 
B. 
C. 
D. 
Answer:
Let 
--------(1) (∵ α is angle between a⃗ and b⃗ )
Then 
(∵θ is angle between e⃗ and c⃗ ⇒ θ is angle between a⃗ Xb⃗ and c⃗ )
(∵ using (1))
Hence, 
if and only if 
if and only if 
and 
if and only if 
and 
⇒ a⃗ and b⃗ are perpendicular.
Also e⃗ is perpendicular to both a⃗ and b⃗ .
θ=0⇒ c⃗ is perpendicular to both a⃗ and b⃗
∴ a⃗, b⃗, c⃗ are mutually perpendicular.
∴ a⃗∙ b⃗=b⃗∙ c⃗=c⃗∙ a⃗=0
Question 13.
Mark the correct alternative in each of the following:
A. 0
B. 
C.
D.
Answer:
Question 14.
Mark the correct alternative in each of the following:
If a⃗ ,b⃗,c⃗ are three non-coplanar vectors, then 
equal.
A. 0
B.
C. 2
D.
Answer:
+ [
]
+ [c⃗ b⃗ a⃗] + 
+ 0
Question 15.
Mark the correct alternative in each of the following:
is equal to
A.
B. 2
C. 3
D. 0
Answer:
+
