Class 12 Maths Chapter 10 Vector Algebra – Exercise 10.5 NCERT Solutions

Introduction

Exercise 10.5 focuses on the scalar triple product of vectors. Students learn how to compute scalar triple products, understand their geometric meaning, and apply them to problems involving volume of parallelepipeds and coplanarity of vectors. This exercise is crucial for advanced applications in vector algebra and 3D geometry.

Formula Used

1. Scalar Triple Product Definition: If $\vec{a},\vec{b},\vec{c}$ are vectors, then

$$\vec{a}\cdot (\vec{b}\times \vec{c})=\mid \begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\mid$$

2. Geometric Meaning: $\mathrm{\mid }\vec{a}\cdot (\vec{b}\times \vec{c})\mathrm{\mid }$ = Volume of parallelepiped formed by $\vec{a},\vec{b},\vec{c}$.

3. Coplanarity Condition: If $\vec{a}\cdot (\vec{b}\times \vec{c})=0$, then vectors are coplanar.

Students Frequently Make Mistakes

• Forgetting determinant expansion order.

• Confusing scalar triple product with vector triple product.

• Errors in sign convention.

• Misinterpreting geometric meaning (volume vs area).

• Skipping coplanarity condition.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Find $(\widehat{i},\widehat{j},\widehat{k})$.

$$\widehat{i}\cdot (\widehat{j}\times \widehat{k})=\mid \begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\mid =1$$

Q2. Find $(\widehat{i},\widehat{j},\widehat{j})$.

$$\widehat{i}\cdot (\widehat{j}\times \widehat{j})=0$$

Q3. Find $(\widehat{i}+2\widehat{j},\widehat{j},\widehat{k})$.

$$=\mid \begin{array}{ccc}1& 2& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\mid =1$$

Q4. Find $(2\widehat{i}-\widehat{j}+\widehat{k},\widehat{i}+3\widehat{j}-2\widehat{k},\widehat{i}-\widehat{j}+\widehat{k})$.

$$=\mid \begin{array}{ccc}2& -1& 1\\ 1& 3& -2\\ 1& -1& 1\end{array}\mid =2(3+2)-(-1)(1-(-2))+(1)(-1-3)=10+3-4=9$$

Q5. Find volume of parallelepiped formed by $\vec{a}=\widehat{i}+2\widehat{j}+3\widehat{k},\vec{b}=2\widehat{i}+3\widehat{j}+4\widehat{k},\vec{c}=3\widehat{i}+4\widehat{j}+5\widehat{k}$.

$$=\mid \begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 3& 4& 5\end{array}\mid =0$$

Vectors are coplanar ⇒ Volume = 0.

Q6. Find $(\widehat{i}+2\widehat{j}+3\widehat{k},\widehat{i},\widehat{j})$.

$$=\mid \begin{array}{ccc}1& 2& 3\\ 1& 0& 0\\ 0& 1& 0\end{array}\mid =1$$

Q7. Find $(\widehat{i}+2\widehat{j},\widehat{j}+3\widehat{k},\widehat{k})$.

$$=\mid \begin{array}{ccc}1& 2& 0\\ 0& 1& 3\\ 0& 0& 1\end{array}\mid =1$$

Q8. Find $(\widehat{i}+2\widehat{j}+3\widehat{k},2\widehat{i}+3\widehat{j}+4\widehat{k},3\widehat{i}+4\widehat{j}+6\widehat{k})$.

$$=\mid \begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 3& 4& 6\end{array}\mid =0$$

Vectors are coplanar.

Q9. Find $(\widehat{i}+2\widehat{j}+3\widehat{k},\widehat{i}-\widehat{j},\widehat{j}-\widehat{k})$.

$$=\mid \begin{array}{ccc}1& 2& 3\\ 1& -1& 0\\ 0& 1& -1\end{array}\mid =1(-1+0)-2(-1-0)+3(1-0)=-1+2+3=4$$

Q10. Find volume of parallelepiped formed by $\vec{a}=\widehat{i},\vec{b}=\widehat{j},\vec{c}=\widehat{k}$.

$$=\mid \begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\mid =1$$

Volume = 1.

FAQs (10)

FAQ1. What is scalar triple product? $\vec{a}\cdot (\vec{b}\times \vec{c})$.

FAQ2. What is geometric meaning of scalar triple product? Volume of parallelepiped.

FAQ3. When are vectors coplanar? If scalar triple product = 0.

FAQ4. What is $(\widehat{i},\widehat{j},\widehat{k})$?

FAQ5. What is $(\widehat{i},\widehat{j},\widehat{j})$?

FAQ6. What is volume of parallelepiped formed by $\widehat{i},\widehat{j},\widehat{k}$?

FAQ7. Why use determinant in scalar triple product? It simplifies computation.

FAQ8. What is coplanarity condition? Scalar triple product = 0.

FAQ9. What is difference between scalar and vector triple product? Scalar triple product gives scalar, vector triple product gives vector.

FAQ10. Why is Exercise 10.5 important? It builds mastery of scalar triple product and coplanarity.

Conclusion

Exercise 10.5 has 10 solved questions and 10 FAQs that strengthen your understanding of scalar triple product and coplanarity of vectors. This completes the Vector Algebra chapter in Class 12 Maths.

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