05_Ex 7.5

Exercise Questions

Questions	Links
Q1. Find the volume of parallelepiped…	7.5 Q-1
Q2. Verify that…	7.5 Q-2
Q3. Prove that the vectors are coplanar…	7.5 Q-3
Q4. Find the $\alpha$ such that the vector is coplanar.	7.5 Q-4
Q5. (a) Find the value of, (b) Prove that…	7.5 Q-5
Q6. Find the values of tetrahedron with the vertices…	7.5 Q-6
Q7. Find the work done, if the point…	7.5 Q-7
Q8. A particle, acted by constant force $4 \underline{\mathbf{i}}+\underline{\mathbf{j}}-3 \underline{\mathbf{k}}$ and…	7.5 Q-8
Q9. A particle is displaced from the point…	7.5 Q-9
Q10. A force of magnitude 6 unit acing parallel…	7.5 Q-10
Q11. A force $\underline{F}=3 \underline{\mathbf{i}}+2 \underline{\mathbf{j}}-4 \underline{\mathbf{k}}$ is applied to…	7.5 Q-11
Q12. A force $\underline{F}=4 \underline{\mathbf{i}}-3 \underline{k}$ passes…	7.5 Q-12
Q13. A force $\underline{F}=\mathbf{2} \underline{\mathbf{i}}+\mathbf{1} \underline{\mathbf{j}}-3 \underline{\mathbf{k}}$ acting…	7.5 Q-13
Q14. Find the moment about $\mathbf{A}(1,1,1)$ of each of…	7.5 Q-14
Q15. A force $\underline{F}=7 \underline{\mathbf{i}}+4 \underline{\mathbf{j}}-\mathbf{3 k}$ is applied…	7.5 Q-15

Overview

This exercise focuses on the concept of scalar triple products and their properties, applications, and computational methods. Scalar triple products are used to calculate volumes of geometric shapes like parallelepipeds and tetrahedrons formed by vectors. These products also help in understanding the relationships between vectors in space, such as co-planarity, and are important in physics, particularly in the study of work and force.

Key Concepts

Scalar Triple Product: This involves three vectors and can be expressed as $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$ . It has significant geometrical and physical applications, especially in calculating volumes and determining the relative orientation of vectors.
Co-planarity: If the scalar triple product of three vectors is zero, the vectors are co-planar, meaning they lie in the same plane.
Volume of Parallelepiped and Tetrahedron: The scalar triple product gives the volume of a parallelepiped formed by three vectors. The volume of a tetrahedron can also be derived using the scalar triple product.
Dot and Cross Products: Fundamental vector operations that are central to understanding the scalar triple product and its applications.

Important Formulas

Scalar Triple Product:

[\mathbf{u} ; \mathbf{v} ; \mathbf{w}] = \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = (\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}

The scalar triple product measures the volume of the parallelepiped formed by the vectors.

Volume of a Parallelepiped:

\text{Volume} = |\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})|

Volume of a Tetrahedron:

\text{Volume} = \frac{1}{6} |\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})|

Zero Scalar Triple Product for Co-planar Vectors:

[\mathbf{u} ; \mathbf{v} ; \mathbf{w}] = 0 \quad \text{if} \quad \mathbf{u}, \mathbf{v}, \mathbf{w} \text{ are co-planar}

Analytical Expression for Scalar Triple Product:

\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = \begin{vmatrix} u_{1} & u_{2} & u_{3} v_{1} & v_{2} & v_{3} w_{1} & w_{2} & w_{3} \end{vmatrix}

Tips and Tricks

Co-planarity Check: To check if three vectors are co-planar, compute their scalar triple product. If it equals zero, the vectors are co-planar.
Right-Hand Rule: For cross products, use the right-hand rule to determine the direction of the resulting vector.
Dot Product of Orthogonal Vectors: The dot product of two orthogonal (perpendicular) vectors is zero. Use this property to check for perpendicularity.
Volume Calculations: When calculating the volume of a tetrahedron, remember to multiply by $\frac{1}{6}$ of the parallelepiped’s volume.
Vector Component Calculations: Break down vectors into components to simplify computations, especially when working with determinants for scalar triple products.

Summary

This exercise covers the scalar triple product, which is crucial for calculating volumes of geometric shapes like parallelepipeds and tetrahedrons formed by three vectors. Key properties include:

The scalar triple product equals zero if the vectors are co-planar.
The volume of a parallelepiped is the absolute value of the scalar triple product.
The scalar triple product can be computed using a determinant when the vectors are given in component form.

Reference

By Sir Shahzad Sair: