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# RD Sharma solutions for Class 12 Mathematics chapter 26 - Scalar Triple Product

## Chapter 26: Scalar Triple Product

#### Chapter 26: Scalar Triple Product Exercise 26.1 solutions [Pages 16 - 17]

Ex. 26.1 | Q 1.1 | Page 16

Evaluate the following:

$\left[\hat{i}\hat{j}\hat{k} \right] + \left[ \hat{j}\hat{k}\hat {i} \right] + \left[ \hat{k}\hat{i} \hat{j} \right]$

Ex. 26.1 | Q 1.2 | Page 16

Evaluate the following:

$\left[ 2 \hat{i}\hat{ j}\ \hat{k}\right] + \left[\hat{i}\hat{ k}\hat {j} \right] + \left[\hat{ k}\hat{ j} 2\hat{ i} \right]$

Ex. 26.1 | Q 2.1 | Page 16

Find $\left[ \vec{a} \vec{b} \vec{c} \right]$ , when $\vec{a} = 2 \hat{i} - 3 \hat{j} , \vec{b} = \hat{i} + \hat{j} - \hat{k} \text{ and } \vec{c} = 3 \hat{i} - \hat{k}$

Ex. 26.1 | Q 2.2 | Page 16

Find $\left[ \vec{a} \vec{b} \vec{c} \right]$ , when $\vec{a} =\hat{ i} - 2 \hat{j} + 3 \hat{k} , \vec{b} = 2 \hat{i} + \hat{j} - \hat{k}\text{ and } \vec{c} = \hat{j} + \hat{k}$

Ex. 26.1 | Q 3.1 | Page 16

Find the volume of the parallelopiped whose coterminous edges are represented by the vector:

$\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} , \vec{b} =\hat{ i} + 2 \hat{j} - \hat{k} , \vec{c} = 3 \hat{i} - \hat{j} + 2 \hat{k}$

Ex. 26.1 | Q 3.2 | Page 16

Find the volume of the parallelopiped whose coterminous edges are represented by the vector:

$\vec{a} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k} , \vec{b} = \hat{i} + 2 \hat{j} - \hat{k} , \vec{c} = 3 \hat{i} - \hat{j} - 2 \hat{k}$

Ex. 26.1 | Q 3.3 | Page 16

Find the volume of the parallelopiped whose coterminous edges are represented by the vector:

$\vec{a} = 11 \hat{i} , \vec{b} = 2 \hat{j} , \vec{c} = 13 \hat{k}$

Ex. 26.1 | Q 3.4 | Page 16

Find the volume of the parallelopiped whose coterminous edges are represented by the vector:

$\vec{a} = \hat{i} + \hat{j} + \hat{k} , \vec{b} =\hat{ i} - \hat{j} + \hat{k} , \vec{c} = \hat{i} + 2 \hat{j} - \hat{k}$

Ex. 26.1 | Q 4.1 | Page 16

Show of the following triad of vector is coplanar:

$\vec{a} = \hat {i} + 2 \hat{j} - \hat {k} , \vec{b} = 3 \hat {i} + 2 \hat{j} + 7 \hat {k} , \vec{c} = 5 \hat {i} + 6 \hat { j} + 5 \hat {k}$

Ex. 26.1 | Q 4.2 | Page 16

Show of the following triad of vector is coplanar:

$\vec{a} = - 4 \hat{i} - 6 \hat{j} - 2 \hat{k} , \vec{b} = -\hat{ i} + 4 \hat{j} + 3 \hat{k} , \vec{c} = - 8 \hat{i} - \hat{j} + 3 \hat{k}$

Ex. 26.1 | Q 4.3 | Page 16

Show of the following triad of vector is coplanar:

$\hat{a} = \hat{i} - 2 \hat {j} + 3 \hat {k} , \hat {b} = - 2 \hat {i} + 3 \hat {j} - 4 \hat { k}, \hat {c} = \hat { i} - 3 \hat { j} + 5 \hat { k }$

Ex. 26.1 | Q 5.1 | Page 16

Find the value of λ so that the following vector is coplanar:

$\vec{a} = \hat{i} - \hat{j} + \hat{k} , \vec{b} = 2 \hat {i} + \hat {j} - \hat {k} , \vec{c} = \lambda\hat { i} - \hat {j} + \lambda \hat {k}$

Ex. 26.1 | Q 5.2 | Page 16

Find the value of λ so that the following vector is coplanar:

$\vec{a} = 2 \hat{i} - \hat {j} + \hat {k} , \vec{b} = \hat {i} + 2 \hat {j} - 3 \hat {k} , \vec{c} = \lambda \hat {i} + \lambda \hat {j} + 5 \hat {k}$

Ex. 26.1 | Q 5.3 | Page 16

Find the value of λ so that the following vector is coplanar:

$\vec{a} = \hat{i} + 2\hat { j} - 3 \hat {k} , \vec{b} = 3 \hat{i} + \lambda \hat {j} + \hat {k} , \vec{c} = \hat {i} + 2 \hat {j} + 2 \hat {k}$

Ex. 26.1 | Q 5.4 | Page 16

Find the value of λ so that the following vector is coplanar:

$\vec{a} = \hat {i} + 3 \hat {j} , \vec{b} = 5 \hat {k} , \vec{c} = \lambda \hat {i} - \hat {j}$

Ex. 26.1 | Q 6 | Page 17

Show that the four points having position vectors

$6 \hat { i} - 7 \hat { j} , 16 \hat {i} - 19 \hat {j}- 4 \hat {k} , 3 \hat {j} - 6 \hat {k} , 2 \hat {i} + 5 \hat {j} + 10 \hat {k}$ are not coplanar.

Ex. 26.1 | Q 7 | Page 17

Show that the points A (−1, 4, −3), B (3, 2, −5), C (−3, 8, −5) and D (−3, 2, 1) are coplanar.

Ex. 26.1 | Q 8 | Page 17

Show that four points whose position vectors are

$6 \hat { i} - 7 \hat {j} , 16 \hat { i} - 19 \hat { j} - 4 \hat {k} , 3 \hat {i} - 6 \hat {k} , 2 \hat { i} - 5 \hat {j}+ 10 \hat {k}$

Ex. 26.1 | Q 9 | Page 17

Find the value of λ for which the four points with position vectors

$-\hat { j} - \hat {k} , 4 \hat {i} + 5 \hat {j} + \lambda \hat {k} , 3 \hat {i} + 9 \hat {j} + 4 \hat {k} \text { and } - 4 \hat {i} + 4 \hat {j} + 4 \hat{k}$

Ex. 26.1 | Q 10 | Page 17

Prove that: $\left( \vec{a} - \vec{b} \right) \cdot \left\{ \left( \vec{b} - \vec{c} \right) \times \left( \vec{c} - \vec{a} \right) \right\} = 0$

Ex. 26.1 | Q 11 | Page 17
$\vec{a,} \vec{b} \text { and } \vec{c}$  are the position vectors of points A, B and C respectively, prove that: $\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}$is a vector perpendicular to the plane of triangle ABC.
Ex. 26.1 | Q 12.1 | Page 17

$\text {Let } \vec{a} = \hat {i} + \hat {j} + \hat {k} , \vec{b} = \hat {i} \text{and} \hat {c} = c_1 \hat{i} + c_2 \hat {j} + c_3 \hat {k} . \text {Then},$

If c1 = 1 and c2 = 2, find c3 which makes $\vec{a,} \vec{b} \text { and } \vec{c}$ coplanar.

Ex. 26.1 | Q 12.2 | Page 17

$\text {Let } \vec{a} = \hat {i} + \hat {j} + \hat {k} , \vec{b} = \hat {i} \text{ and } \hat {c} = c_1 \hat{i} + c_2 \hat {j} + c_3 \hat {k} . \text {Then},$

If c2 = −1 and c3 = 1, show that no value of c1 can make $\vec{a,} \vec{b}\text { and } \vec{c}$ coplanar.

Ex. 26.1 | Q 13 | Page 17

Find λ for which the points A (3, 2, 1), B (4, λ, 5), C (4, 2, −2) and D (6, 5, −1) are coplanar.

Ex. 26.1 | Q 14 | Page 17

If four points A, B, C and D with position vectors 4 $\hat { i} +3$ $\hat { j} +3$ $\hat { k} ,5$ $\hat { i} +$ $x\hat { j} +7$ $\hat { k} ,5$ $\hat { i} +3$ $\hat { j}$ and $7 \hat{i} + 6 \hat{j} + \hat{k}$ respectively are coplanar, then find the value of x.

#### Chapter 26: Scalar Triple Product Exercise 26.1, Very Short Answers solutions [Pages 17 - 18]

Ex. 26.1 | Q 1 | Page 17

Write the value of $\left[ 2 \hat { i } \ 3 \hat { j }\ 4 \hat { k } \right] .$

Ex. 26.1 | Q 2 | Page 17

Write the value of $\left[ \hat {i} + \hat {j} \ \hat {j} + \hat {k} \ \hat {k} + \hat {i} \right] .$

Ex. 26.1 | Q 3 | Page 17

Write the value of $\left[ \hat {i} - \hat {j} \hat {j} - \hat {k} \hat {k} - \hat {i} \right] .$

Ex. 26.1 | Q 4 | Page 17

Find the values of 'a' for which the vectors

$\vec{\alpha} = \hat {i} + 2 \hat {j} + \hat {k} , \vec{\beta} = a \hat {i} + \hat {j} + 2 \hat {k} \text { and } \vec{\gamma} = \hat {i} + 2 \hat {j} + a \hat { k }$ are coplanar.

Very Short Answers | Q 5 | Page 18

Find the volume of the parallelopiped with its edges represented by the vectors $\hat {i} + \hat {j} , \hat {i} + 2 \hat {j} \text { and } \hat {i} + \hat {j} + \pi k .$

Very Short Answers | Q 6 | Page 18

If $\vec{a,} \vec{b}$ $\text { are non-collinear vectors, then find the value of} \left[ \vec{a} \vec{b}\hat { i} \right] \hat{i} + \left[ \vec{a} \vec{b} \hat {j} \right] \hat {j} + \left[ \vec{a} \vec{b} \hat {k} \right] \hat {k} .$

Very Short Answers | Q 7 | Page 18

If the vectors (sec2 A) $\hat {i} + \hat {j} + \hat {k} , \hat {i} + \left( \sec^2 B \right) \hat {j} + \hat {k} , \hat {i} + \hat {j} + \left( \sec^2 C \right) \hat {k}$ are coplanar, then find the value of cosec2 A + cosec2 B + cosec2 C.

Very Short Answers | Q 8 | Page 18

For any two vectors $\vec{a} \text { and } \vec{b}$ of magnitudes 3 and 4 respectively, write the value of $\left[ \vec{a} \vec{b} \vec{a} \times \vec{b} \right] + \left( \vec{a} \cdot \vec{b} \right)^2 .$

Very Short Answers | Q 9 | Page 18

If $\left[ 3 \vec{a} + 7 \vec{b} \vec{c} \vec{d} \right] = \lambda\left[ \vec{a} \vec{c} \vec{d} \right] + \mu\left[ \vec{b} \vec{c} \vec{d} \right],$ then find the value of λ + μ.

Very Short Answers | Q 10 | Page 18

If $\vec{a,} \vec{b,} \vec{c}$ are non-coplanar vectors, then find the value of $\frac{\vec{a} \cdot \left( \vec{b} \times \vec{c} \right)}{\left( \vec{c} \times \vec{a} \right) \cdot \vec{b}} + \frac{\vec{b} \cdot \left( \vec{a} \times \vec{c} \right)}{\vec{c} \cdot \left( \vec{a} \times \vec{b} \right)}$.

Very Short Answers | Q 11 | Page 18

Find $\vec{a} . \left( \vec{b} \times \vec{c} \right)$,  if $\vec{a} = 2 \hat {i} + \hat {j} + 3 \hat {k} , \vec{b} = - \hat {i} + 2 \hat {j} + \hat {k}$ and  $\vec{c} = 3 \hat { i} + \hat {j} + 2 \hat {k}$.

#### Chapter 26: Scalar Triple Product Exercise MCQ solutions [Pages 18 - 20]

MCQ | Q 1 | Page 18

If $\vec{a}$ lies in the plane of vectors $\vec{b} \text { and } \vec{c}$, then which of the following is correct?

• $\left[ \vec{a} \vec{b} \vec{c} \right] = 0$

• $\left[ \vec{a} \vec{b} \vec{c} \right] = 1$

• $\left[ \vec{a} \vec{b} \vec{c} \right] = 3$

• $\left[ \vec{b} \vec{c} \vec{a} \right] = 1$

MCQ | Q 2 | Page 18

The value of $\left[ \vec{a} - \vec{b} , \vec{b} - \vec{c} , \vec{c} - \vec{a} \right], \text { where } \left| \vec{a} \right| = 1, \left| \vec{b} \right| = 5, \left| \vec{c} \right| = 3, \text { is }$

• 0

• 1

• 6

• none of these

MCQ | Q 3 | Page 18

If $\vec{a,} \vec{b,} \vec{c}$ are three non-coplanar mutually perpendicular unit vectors, then $\left[ \vec{a} \vec{b} \vec{c} \right],$ is

• ± 1

• 0

• -2

• 2

MCQ | Q 4 | Page 18

If $\vec{r} \cdot \vec{a} = \vec{r} \cdot \vec{b} = \vec{r} \cdot \vec{c} = 0$ for some non-zero vector $\vec{r} ,$ then the value of $\left[ \vec{a} \vec{b} \vec{c} \right],$ is

• 2

• 3

• 0

• none of these

MCQ | Q 5 | Page 19

For any three vectors $\vec{a,} \vec{b,} \vec{c}$  the expression $\left( \vec{a} - \vec{b} \right) . \left\{ \left( \vec{b} - \vec{c} \right) \times \left( \vec{c} - \vec{a} \right) \right\}$  equals

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

• $2\left[ \vec{a} \vec{b} \vec{c} \right]$

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

• none of these

MCQ | Q 6 | Page 19

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

• 0

• 2

• 1

• none of these

MCQ | Q 7 | Page 19

Let $\vec{a} = a_1 \hat { i }+ a_2 \hat {j} + a_3 \hat {k} , \vec{b} = b_1 \hat {i} + b_2 \hat { j } + b_3 \hat { k} \text { and } \vec{c} = c_1 \hat { i } + c_2 \hat{j } + c_3\text { k }$ be three non-zero vectors such that $\vec{c}$ is a unit vector perpendicular to both $\vec{a} \text { and } \vec{b}$. If the angle between $\vec{a} \text { and } \vec{b}$ is $\frac{\pi}{6},$ , then

$\begin{vmatrix}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{vmatrix}^2$ is equal to

• 0

• 1

• $\frac{1}{4} \left| \vec{a} \right|^2 \left| \vec{b} \right|^2$

• $\frac{3}{4} \left| \vec{a} \right|^2 \left| \vec{b} \right|^2$

MCQ | Q 8 | Page 19

If $\vec{a} = 2\hat{ i} - 3 \hat { j} + 5 \hat { k} , \vec{b} = 3 \hat {i} - 4 \hat {j} + 5 \hat {k} \text { and } \vec{c} = 5\hat { i } - 3 \hat {j}- 2 \hat{k},$ then the volume of the parallelopiped with conterminous edges $\vec{a} + \vec{b,} \vec{b} + \vec{c,} \vec{c} + \vec{a}$ is

• 2

• 1

• -1

• 0

• None of these

MCQ | Q 9 | Page 19

If $\left[ 2 \vec{a} + 4 \vec{b} \vec{c} \vec{d} \right] = \lambda\left[ \vec{a} \vec{c} \vec{d} \right] + \mu\left[ \vec{b} \vec{c} \vec{d} \right],$  then λ + μ =

• 6

• -6

• 10

• 8

MCQ | Q 10 | Page 19

$\left[ \vec{a} \vec{b} \vec{a} \times \vec{b} \right] + \left( \vec{a} . \vec{b} \right)^2 =$

• $\left| \vec{a} \right|^2 \left| \vec{b} \right|^2$

• $\left| \vec{a} + \vec{b} \right|^2$

• $\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2$

• $2 \left| \vec{a} \right|^2 \left| \vec{b} \right|^2$

MCQ | Q 11 | Page 19

If the vectors $4 \hat { i} + 11 \hat {j} + m \hat {k} , 7 \hat { i} + 2 \hat { j} + 6 \hat {k} \text { and } \hat {i} + 5 \hat {j} + 4 \hat {k}$ are coplanar, then m =

• 0

• 38

• -10

• 10

MCQ | Q 12 | Page 19

For non-zero vectors $\vec{a,} \vec{b} \text { and }\vec{c}$ the relation $\left| \left( \vec{a} \times \vec{b} \right) \cdot \vec{c} \right| = \left| \vec{a} \right| \left| \vec{b} \right| \left| \vec{c} \right|$ holds good, if

• $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = 0$

• $\vec{a} \cdot \vec{b} = 0 = \vec{c} \cdot \vec{a}$

• $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = 0$

• $\vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = 0$

MCQ | Q 13 | Page 19

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

• 0

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

• $2\left[ \vec{a} \vec{b} \vec{c} \right]$

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

MCQ | Q 14 | Page 19

If $\vec{a,} \vec{b,} \vec{c}$ are three non-coplanar vectors, then $\left( \vec{a} + \vec{b} + \vec{c} \right) . \left[ \left( \vec{a} + \vec{b} \right) \times \left( \vec{a} + \vec{c} \right) \right]$ equals

• 0

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

• $2\left[ \vec{a} \vec{b} \vec{c} \right]$

• $- \left[ \vec{a} \vec{b} \vec{c} \right]$

MCQ | Q 15 | Page 20

$\left( \vec{a} + 2 \vec{b} - \vec{c} \right) \cdot \left\{ \left( \vec{a} - \vec{b} \right) \times \left( \vec{a} - \vec{b} - \vec{c} \right) \right\}$ is equal to

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

• $2\left[ \vec{a} \vec{b} \vec{c} \right]$

• $3\left[ \vec{a} \vec{b} \vec{c} \right]$

• 0

## RD Sharma solutions for Class 12 Mathematics chapter 26 - Scalar Triple Product

RD Sharma solutions for Class 12 Maths chapter 26 (Scalar Triple Product) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 12 Mathematics chapter 26 Scalar Triple Product are Concept of Direction Cosines, Properties of Vector Addition, Geometrical Interpretation of Scalar, Scalar Triple Product of Vectors, Vector (Or Cross) Product of Two Vectors, Scalar (Or Dot) Product of Two Vectors, Position Vector of a Point Dividing a Line Segment in a Given Ratio, Multiplication of a Vector by a Scalar, Addition of Vectors, Introduction of Vector, Magnitude and Direction of a Vector, Basic Concepts of Vector Algebra, Types of Vectors, Components of a Vector, Section formula, Vector Joining Two Points, Vectors Examples and Solutions, Projection of a Vector on a Line, Introduction of Product of Two Vectors.

Using RD Sharma Class 12 solutions Scalar Triple Product exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer RD Sharma Textbook Solutions to score more in exam.

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