For any three vectors ${\bf{\vec a}},{\bf{\vec b}}$ and ${{\bf{\vec c}}}$, $\left( {{\bf{\vec a}} - {\bf{\vec b}}} \right)\cdot\left( {{\bf{\vec b}} + {\bf{\vec c}}\;} \right) \times \left( {\;{\bf{\vec c}} + {\bf{\vec a}}} \right)$ is equal to

$2{\bf{\vec a}}\cdot\left( {{\bf{\vec b}} \times {\bf{\vec c}}\;} \right)$

$\left[ {\;{\bf{\vec a}}\;{\bf{\vec b}}\;{\bf{\vec c}}\;} \right]$

${\left[ {\;{\bf{\vec a}}\;{\bf{\vec b}}\;{\bf{\vec c}}\;} \right]^2}$

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JEE • MATHEMATICS • SCALAR TRIPLE PRODUCT

Scalar triple product practice · Mixed difficulty

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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91 Easy

91 of 230

Application +4 | -1

If the volume of parallelopiped formed by the vectors

\hat i + \lambda \hat j + \hat k,\hat j + \lambda \hat k

and

\lambda \hat i + \hat k

is minimum, then

\lambda

is equal to

View answer and explanation

Correct option: B

\frac{1}{{\sqrt 3 }}

Option B (

\frac{{1}}{{\sqrt 3 }}

) is correct because it is the value of

\lambda

that minimizes the volume of the parallelepiped formed by the given vectors.

92 Easy

92 of 230

Analyzing +4 | -1

\vec{a}, \vec{b}, \vec{c}

are non-coplanar vectors and

\lambda

is a real number, then the vectors

\vec{a}+2 \vec{b}+3 \vec{c}, \lambda \vec{b}+4 \vec{c}

and

(2 \lambda-1) \vec{c}

are non-coplanar for

View answer and explanation

Correct option: C

all except two values of

\lambda

Option C (all except two values of

\lambda

) is correct because the vectors are non-coplanar when the scalar triple product is non-zero, which occurs for all values of

\lambda

except 0 and 1/2.

93 Medium

93 of 230

Analyzing +4 | -1

\alpha {\text{ }}\,\,{\text{and}}\,\,{\text{ }}\beta

are two mutually perpendicular unit vectors

\left\{ {r\alpha + r\beta + s(\alpha \times \beta } \right\},\,\,\,[\alpha + (\alpha \times \beta )]

and

\left\{ {s\alpha + s\beta + t(\alpha \times \beta )} \right\}

are coplanar, then s is equal to

View answer and explanation

Correct option: B

GM of r and t

Option B (GM of r and t) is correct because the coplanarity condition leads to the equation s² = rt, which defines the geometric mean.

94 Easy

94 of 230

Analyzing +4 | -1

\mathbf{u}, \mathbf{v}

and

\mathbf{w}

are non-coplanar vectors and

p, q

are real numbers, then the equality

[3 \mathbf{\,u\,} p\, \mathbf{v\,} p \mathbf{\,w}]-[p\, \mathbf{v\,} \mathbf{w\,} q\, u]

- [2w qv qu]

=0

holds for

View answer and explanation

Correct option: D

exactly one value of

(p, q)

Option D (exactly one value of

(p, q)

) is correct because the equation simplifies to

3p^2 - pq + 2q^2 = 0

, which has only one solution:

p = q = 0

95 Easy

95 of 230

Application +4 | -1

Let the vectors

\vec{a}, \vec{b}, \vec{c}

represent three coterminous edges of a parallelopiped of volume

\text{V}

. Then the volume of the parallelopiped, whose coterminous edges are represented by

\vec{a}, \vec{b}+\vec{c}

and

\vec{a}+2 \vec{b}+3 \vec{c}

is equal to

View answer and explanation

Correct option: C

\text{V}

Option C (V) is correct because the volume of the parallelepiped is given by the scalar triple product, which simplifies to V after expansion.

96 Easy

96 of 230

Application +4 | -1

\vec a,\vec b,\vec c

are non-coplanar vectors then

\frac{{\vec a\cdot\left( {\vec b \times \vec c} \right)}}{{\left( {\vec c \times \vec a} \right)\cdot\vec b}} + \frac{{\vec b\cdot\left( {\vec a \times \vec c} \right)}}{{\vec c\cdot\left( {\vec a \times \vec b} \right)}}

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because the expression simplifies to 1 - 1 = 0 using properties of scalar triple product.

97 Easy

97 of 230

Application +4 | -1

\mathbf{\vec a}\cdot\widehat \mathbf{i} = 4

then

\left( {\mathbf{\vec a} \times \widehat \mathbf{j}} \right)\cdot\left( {2\widehat \mathbf{j} - 3\mathbf{\hat k}} \right)

is equal to

View answer and explanation

Correct option: D

- 12

Option D (

- 12

) is correct because it is the result of the scalar triple product calculation using the given dot product.

98 Easy

98 of 230

Application +4 | -1

Shortest distance between lines:

\frac{x-1}{2}=\frac{2 y-2}{3}=\frac{z-3}{1}

and

\frac{x-2}{3}=\frac{y-1}{2}=\frac{z+2}{4}

View answer and explanation

Correct option: A

\frac{13}{\sqrt{165}}

Option A (

1+\frac{e^2}{4}

) is correct because it satisfies the given differential equation and initial condition when solving for y(e).

99 Easy

99 of 230

Application +4 | -1

The shortest distance between the lines

\frac{{x - 3}}{3} = \frac{{y - 8}}{{ - 1}} = \frac{{z - 3}}{1}

and

\frac{{x + 3}}{{ - 3}} = \frac{{y + 7}}{2} = \frac{{z - 6}}{4}

View answer and explanation

Correct option: A

3\sqrt {30}

Option A (

3\sqrt {30}

) is correct because it is the result of applying the shortest distance formula between two skew lines.

100 Easy

100 of 230

Application +4 | -1

For any three vectors,

\vec{a}, \vec{b}\,\, \vec{c}

, the value of

[\vec{a}+\vec{b} \quad \vec{b}+\vec{c} \,\,\vec{c}+\vec{a}]

, is

View answer and explanation

Correct option: B

2\left[\begin{array}{lll}\vec{a} & \vec{b} & \vec{c}\end{array}\right]

Option B (

2\left[\begin{array}{lll}\vec{a} & \vec{b} & \vec{c}\end{array}\right]

) is correct because the scalar triple product can be expanded and simplified using determinant properties.

101 Medium

101 of 230

Application +4 | -1

Given vectors

\vec x = 3\hat i - 6\hat j - \hat k,\;\vec y = \hat i + 4\hat j - 3\hat k

and

\vec z = 3\hat i + 4\hat j + 12\hat k

, then the projection of

\vec x \times \vec y

on vector

{\vec z}

View answer and explanation

Correct option: B

- 14

Option B (

-14

) is correct because the projection is calculated using the scalar triple product divided by the magnitude of vector z.

102 Easy

102 of 230

Analyzing +4 | -1

Let

G_1, G_2, G_3

be the centroids of the triangular faces

O B C, O C A, O A B

of a tetrahedron

O A B C

. If

V_1

denote the volume of the tetrahedron

\mathrm{OABC}

and

V_2

that of the parallelopiped with

O G_1, O G_2, O G_3

as three concurrent edges, then

View answer and explanation

Correct option: A

4 V_1=9 V_2

Option A (

4 V_1=9 V_2

) is correct because it relates the volume of the tetrahedron to the volume of the parallelepiped formed by the centroids of its faces.

103 Easy

103 of 230

Analyzing +4 | -1

For three vectors

\vec{u}, \vec{v}, \vec{w}

which of the following expressions is not equal to any of the remaining three?

View answer and explanation

Correct option: C

\vec{v} \cdot(\vec{u} \times \vec{w})

Option C (

\vec{v} \cdot(\vec{u} \times \vec{w})

) is correct because it is the only expression that is not equal to the other three due to the properties of the scalar triple product.

104 Easy

104 of 230

Application +4 | -1

The value of scalar triple product

\mathbf{i}-2 \mathbf{j}+3 \mathbf{k}, 2 \mathbf{i}+\mathbf{j}-\mathbf{k}

and

\mathbf{j}+\mathbf{k}

View answer and explanation

Correct option: A

Option A (12) is correct because the scalar triple product is calculated correctly using the determinant of the matrix formed by the components of the vectors.

105 Hard

105 of 230

Application +4 | -1

A vector

\mathbf{n}

of magnitude 8 units is inclined to

x

-axis at

45^{\circ}, y

-axis at

60^{\circ}

and an acute angle with

z

-axis. If a plane passes through a point

(\sqrt{2},-1,1)

and is normal to

\mathbf{n}

, then its equation in vector form is

View answer and explanation

Correct option: B

\mathbf{r} \cdot(\sqrt{2} \mathbf{i}+\mathbf{j}+\mathbf{k})=2

Option B (

\mathbf{r} \cdot(\sqrt{2} \mathbf{i}+\mathbf{j}+\mathbf{k})=2

) is correct because it is the equation of the plane passing through the given point and normal to the vector

\mathbf{n}

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

106 Medium

106 of 230

Application +4 | -1

The shortest distance between the lines

\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}

and

\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3}

View answer and explanation

Correct option: A

4 \sqrt{3}

Option A (

4 \sqrt{3}

) is correct because it is the shortest distance calculated using the formula for skew lines.

107 Easy

107 of 230

Understanding +4 | -1

\frac{{\mathbf{\vec a}\cdot\left( {\;\mathbf{\vec b} \times \;\mathbf{\vec c}} \right)}}{{\mathbf{\vec b}\cdot\left( {\;\mathbf{\vec c} \times \mathbf{\vec a}} \right)}} + \frac{{\mathbf{\vec b}\cdot\left( {\;\mathbf{\vec a} \times \mathbf{\vec b}} \right)}}{{\mathbf{\vec a}\cdot\left( {\mathbf{\vec b} \times \;\mathbf{\vec c}} \right)}}\;\mathrm{is\;equal\;to}

View answer and explanation

Correct option: A

Option A (1) is correct because the expression simplifies to 1 using properties of scalar triple products.

108 Medium

108 of 230

Application +4 | -1

Equation of the plane through three points

A, B

and

C

with position vectors

-6 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k}, 3 \mathbf{i}-2 \mathbf{j}+4 \mathbf{k}

and

5 \mathbf{i}+7 \mathbf{j}+3 \mathbf{k}

is equal to

View answer and explanation

Correct option: A

\mathbf{r} \cdot(\mathbf{i}-\mathbf{j}+7 \mathbf{k})+23=0

Option A (

\mathbf{r} \cdot(\mathbf{i}-\mathbf{j}+7 \mathbf{k})+23=0

) is correct because it represents the equation of the plane passing through the three given points.

109 Easy

109 of 230

Application +4 | -1

\mathbf{a}, \mathbf{b}

and

\mathbf{c}

are non-coplanar vectors and

\mathbf{d}=\lambda \mathbf{a}+\mu \mathbf{b}+v \mathbf{c}

, then

\lambda

is equal to

View answer and explanation

Correct option: B

\frac{\text { [bcd] }}{\text { [bca] }}

Option B (

\frac{\text { [bcd] }}{\text { [bca] }}

) is correct because by taking the dot product of

\mathbf{d}

with

\mathbf{b} \times \mathbf{c}

, we can isolate

\lambda

and express it in terms of scalar triple products.

110 Easy

110 of 230

Application +4 | -1

\vec{a}, \vec{b}, \vec{c}

are three non-coplanar vectors represented by non-current edges of a parallelopiped of volume 4 units, then the value of

(\vec a + \vec b) \cdot (\vec b \times \vec c) + (\vec b + \vec c) \cdot (\vec c \times \vec a) + (\vec c + \vec a) \cdot (\vec a \times \vec b){\text{, is }}

View answer and explanation

Correct option: C

\pm 12

Option C (

\pm 12

) is correct because the expression simplifies to three times the scalar triple product, which is related to the volume of the parallelepiped.

111 Medium

111 of 230

Analyzing +4 | -1

Let

\vec{a}=4 \hat{i}+3 \hat{j}

and

\vec{b}=3 \hat{i}-4 \hat{j}+5 \hat{k}

. If

\vec{c}

is a vector such that

\vec{c} \cdot(\vec{a} \times \vec{b})+25=0, \vec{c} \cdot(\hat{i}+\hat{j}+\hat{k})=4

and projection of

\vec{c}

\vec{a}

is 1 , then the projection of

\vec{c}

\vec{b}

equals

View answer and explanation

Correct option: B

\frac{5}{\sqrt{2}}

Option B (

\frac{5}{\sqrt{2}}

) is correct because it is the value obtained by correctly solving the system of equations and then calculating the projection of c on b.

112 Easy

112 of 230

Application +4 | -1

The value of

a

so that the volume of parallelopiped formed by

\widehat \mathbf{i} + a\widehat \mathbf{j} + \mathbf{\hat k},\;\widehat \mathbf{j} + a\mathbf{\hat k}

and

a\widehat \mathbf{i} + \mathbf{\hat k}

becomes minimum is

View answer and explanation

Correct option: C

1/\sqrt 3

Option C (

1/\sqrt 3

) is correct because it minimizes the volume of the parallelepiped, which is found by taking the derivative and setting it to zero.

113 Easy

113 of 230

Application +4 | -1

\mathbf{a} \cdot \mathbf{i}=4

, then

(a \times \mathbf{j}) \cdot(2 \mathbf{j}-3 \mathbf{k})=

View answer and explanation

Correct option: D

-12

Option D (-12) is correct because the expression (a x j) . (2j - 3k) simplifies to -3(a . i), and given a . i = 4, the result is -12.

114 Easy

114 of 230

Application +4 | -1

The length of the shortest distance between the two lines

\mathbf{\vec r} = \left( { - 3\widehat \mathbf{i} + 6\widehat \mathbf{j}} \right) + s\left( { - 4\widehat \mathbf{i} + 3\widehat \mathbf{j} + 2\mathbf{\hat k}} \right)

and

\mathbf{\vec r} = \left( { - 2\widehat \mathbf{i} + 7\mathbf{\hat k}} \right) + t\left( { - 4\widehat \mathbf{i} + \widehat \mathbf{j} + \mathbf{\hat k}} \right)\mathrm{\;}

View answer and explanation

Correct option: D

9 units

Option D (9 units) is correct because the shortest distance between the two lines is calculated using the formula involving the vector difference and the cross product of the direction vectors.

115 Easy

115 of 230

Analyzing +4 | -1

For any three vectors

\mathbf{\vec a},\mathbf{\vec b}

and

{\mathbf{\vec c}}

\left( {\mathbf{\vec a} - \mathbf{\vec b}} \right)\cdot\left( {\mathbf{\vec b} + \mathbf{\vec c}\;} \right) \times \left( {\;\mathbf{\vec c} + \mathbf{\vec a}} \right)

is equal to

View answer and explanation

Correct option: D

Option D (0) is correct because the scalar triple product simplifies to zero due to vector properties and cancellations.

116 Easy

116 of 230

Application +4 | -1

If the volume of the parallelopiped with

\mathbf{\vec a},\;\mathbf{\vec b}

and

{\mathbf{\vec c}}

as coterminous edges is 40 cu units, then the volume of the parallelopiped having

\mathbf{\vec b} + \mathbf{\vec c},\;\mathbf{\vec c} + \mathbf{\vec a}

and

\mathbf{\vec a} + \mathbf{\vec b}

as coterminous edges inn cubic units is

View answer and explanation

Correct option: A

Option A (80) is correct because the volume of the new parallelepiped is twice the volume of the original parallelepiped.

117 Easy

117 of 230

Application +4 | -1

The shortest distance between the lines whose equations are

\vec{r}=t(\hat{i}+\hat{j}+\hat{k})

and

\vec{r}=\hat{k}+s(\hat{i}-2 \hat{j}+3

\hat{\mathrm{k}})

is :

View answer and explanation

Correct option: B

\frac{3}{\sqrt{38}}

Option B (

\frac{3}{\sqrt{38}}

) is correct. The shortest distance between two lines is given by the formula involving the cross product of the direction vectors and the vector connecting a point on each line.

118 Easy

118 of 230

Analyzing +4 | -1

\vec{a}, \vec{b}, \vec{c}

are three non-coplanar, non-zero vectors, then

(\vec{a} \cdot \vec{a})(\vec{b} \times \vec{c})+(\vec{a} \cdot \vec{b})(\vec{c} \times \vec{a})+(\vec{a} \cdot \vec{c})(\vec{a} \times \vec{b})

is equal to

View answer and explanation

Correct option: B

\left[\begin{array}{lll}\vec{b} & \vec{c} & \vec{a}\end{array}\right] \vec{a}

Option B (

\left[\begin{array}{lll}\vec{b} & \vec{c} & \vec{a}\end{array}\right] \vec{a}

) is correct because the given expression simplifies to the scalar triple product multiplied by the vector

\vec{a}

119 Easy

119 of 230

Application +4 | -1

Volume of the tetrahedron with vertices

P(-1,2,0), Q(2,1,-3), R(1,0,1)

and

S(3,-2,3)

View answer and explanation

Correct option: B

2 / 3

Option B (2/3) is correct because the volume of the tetrahedron is calculated using the scalar triple product of the vectors formed by the vertices.

120 Medium

120 of 230

Application +4 | -1

Let

\bar{u}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \bar{v}=3 \hat{i}+4 \hat{j}+5 \hat{k}

and

\bar{w}

be a unit vector, then the maximum value of the scalar triple product

[\bar{u} \bar{v} \bar{w}]

, is

View answer and explanation

Correct option: C

\sqrt{6}

Option C (

\sqrt{6}

) is correct because the maximum value of the scalar triple product occurs when

\bar{w}

is in the direction of

\bar{u} \times \bar{v}

, and its magnitude is

|\bar{u} \times \bar{v}|

Revision notes and key takeaways

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