# RS Aggarwal Class 12 Chapter 25 Solutions (Product of Three Vectors)

RS Aggarwal Solutions for Class 12 Chapter 25 ‘Product of Three Vectors’ will help ace your Board exams and other competitive exams. It contains a wide variety of questions to enhance your accuracy and precision during the exam time. This chapter is more or less an application of the cross product and scalar product that you learnt in the previous chapters. You will learn how the product of three vectors can help you in finding the volume of a parallelepiped and lastly, you will be taught the properties of the scalar triple product in the exercise solutions of this chapter.

There are 2 exercises given in this chapter, that consist of 24 questions in total. The questions in this chapter are arranged such that the complexity of the concepts will increase as you will proceed in the exercise questions. You can rest assured once you have completed practising these solutions thoroughly because these questions cover all the important topics in your syllabus. The RS Aggarwal Class 12 Maths Solutions Product of Three Vectors will facilitate the process of learning the application of vectors and their products in a comprehensive manner.

If you are looking for a resource that will cover every aspect of your syllabus in this chapter, Instasolv should be your go-to destination. We have compiled these answers to match the level of your understanding besides helping you build a strong aptitude for approaching the maths problems in your competitive exams. Using Instasolv as your source of reference will help you optimize your self-study hours to the most.

## Summary of Important Topics Covered in RS Aggarwal Solutions for Class 12 Chapter 25 – Product of Three Vectors

**Introduction to the Product of Three Vectors**

The Product of three vectors is also, often termed as the scalar triple product, and as the name suggests, it is given by the multiplying three vectors. To evaluate the scalar triple product, you should calculate the dot product of one vector with the vector product of the other two vectors. It can be mathematically represented as,

[**a b c**]= (**a **X **b**) **. c**

You must note that the following points must be remembered while solving the questions related to the scalar triple product:

- The product of three vectors will always have a resultant that is a scalar quantity.
- The scalar triple product is calculated keeping in mind the following order.

- First, calculate the cross product of two given vectors.
- Then, take the dot product of the third product with the resultant vector.

**Physical Representation of Three Vectors**

The product of three vectors will help you in finding the volume of a parallelepiped of which the three sides with a common corner are represented by the three given vectors **a**, **b **and **c**.

For instance, if we consider the parallelogram with two adjacent sides as vector **a **and **b** with the angle between them representing the base of a parallelepiped and vector **c** giving the height of the same. If the angle between the resultant vector of the cross product of **a **and **b, **and vector **c** is then

Volume of the Parallelepiped = (α X b)c.cosα

⇒ Volume of the Parallelepiped = (α X b).c

**Properties of the Product of Three Vectors**

- In the case of cyclical permutation, (
**a**X**b**)**. c**=**a.**(**b**X**c**) - A cyclic product in nature means,
**[a b c]= [b c a]= [c a b]= -[b a c]= -[a c b]= -[c b a]** **To prove that any three products are coplanar in nature, you must check their scalar triple product. It is the case because if the angle between the resultant vector of the cross product of two vectors with the third vector is 90° then**cos 90°= 0 which would make the scalar triple product equivalent to 0 which can be shown as (**a**X**b**)**. c**= (a X b)*c.cos*90° =0

**Highlights of Product of Three Vectors**

- [
**a b c**]= (**a**X**b**)**. c** - Volume of the Parallelepiped=(a X b).c
- A cyclic product in nature means,

**[a b c]= [b c a]= [c a b]= -[b a c]= -[a c b]= -[c b a]**

### Exercise Discussion of RS Aggarwal Solutions for Class 12 Chapter 25 – Product of Three Vectors

- The exercise solutions of 25A mostly consist of the questions related to checking if the given vectors are parallel or coplanar, which will require you to deal with evaluating the scalar triple product and also, their determinant form.
- The exercise 25B is a level up consisting of questions that will require you to have in-depth knowledge of all the concepts of vectors.
- These questions will train you comprehensively with respect to figuring out the right approach to any math problem related to this topic.
- You must practice these questions besides solving your course textbook to score a good rank in the competitive exams.

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The RS Aggarwal solutions at Instasolv are mentioned exercise-wise so that you can access them easily and solve your queries in minutes. Our maths experts employ research-based analysis in providing you with the answers. We are updated with the latest exam trends and syllabus of all the competitive exams at your level which is why you must rely on us. We are here 24/7 to solve your queries.