02_Ex 7.2

Exercise Questions

Questions	Links
Q1. Let $A(2,5), B(-1,1)$ and $C(2,-6)$ find…	7.2 Q-1
Q2. Let $\underline{U}=\underline{\mathbf{i}}-2 \underline{\mathbf{j}}-\underline{\mathbf{k}},$ …	7.2 Q-2
Q3. Find the magnitude of the vector V and…	7.2 Q-3
Q4. Find $\alpha$ , so that $	7.2 Q-4
Q5. Find a unit vector in the direction of…	7.2 Q-5
Q6. If $\underline{\mathbf{a}}=3 \underline{\mathbf{i}}-\underline{j}-4 \underline{\mathbf{k}}, \underline{\mathbf{b}}=-2 \underline{\mathbf{i}}-4 \underline{\mathrm{j}}-3 \underline{\mathrm{k}}$ and…	7.2 Q-6
Q7. Find a vector whose…	7.2 Q-7
Q8. If $\underline{U}=\mathbf{2} \underline{\mathbf{i}}+3 \underline{\mathbf{j}}+\mathbf{4} \underline{\mathbf{k}}, \underline{\mathrm{V}}=-\underline{\mathbf{i}}+3 \underline{\mathbf{j}}-\underline{\mathrm{k}}$ and…	7.2 Q-8
Q9. The position vectors of the point $A, B, C$ and $D$ are..	7.2 Q-9
Q10. We say that two vectors $\underline{\mathrm{V}}$ and $\underline{\mathrm{W}}$ in space…	7.2 Q-10
Q11. Find the direction cosines for the given…	7.2 Q-11
Q12. Which of the following triples can be…	7.2 Q-12

Overview

This exercise focuses on key concepts in vector analysis, including direction cosines, magnitude, unit vectors, and vector addition. These concepts are fundamental in understanding vector operations in three-dimensional space. The exercise covers the derivation of direction cosines and explores practical applications such as calculating vector magnitudes, vector addition, and determining unit vectors. These principles are essential for various fields, including physics, engineering, and computer graphics.

Key Concepts

Magnitude of a Vector: The magnitude of a vector $\mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$ is given by: $|\mathbf{v}| = \sqrt{a^2 + b^2 + c^2}$
Unit Vector: A unit vector in the direction of a vector $\mathbf{v}$ is obtained by dividing the vector by its magnitude: $\hat{u} = \frac{\mathbf{v}}{|\mathbf{v}|}$
Direction Cosines: These are the cosines of the angles that a vector makes with the coordinate axes. For a vector $\mathbf{v} = (v_x, v_y, v_z)$ , the direction cosines are: $\cos \alpha = \frac{v_x}{|\mathbf{v}|}, \quad \cos \beta = \frac{v_y}{|\mathbf{v}|}, \quad \cos \gamma = \frac{v_z}{|\mathbf{v}|}$
Vector Addition and Subtraction: Vectors can be added or subtracted geometrically, and this concept is crucial for solving problems in vector geometry.

Important Formulas

Magnitude of a Vector: $|\mathbf{v}| = \sqrt{a^2 + b^2 + c^2}$
Unit Vector: $\hat{u} = \frac{\mathbf{v}}{|\mathbf{v}|}$
Direction Cosines: $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$
Vector Addition (Triangle Law): If vectors $\mathbf{u}$ , $\mathbf{v}$ , and $\mathbf{w}$ represent the sides of a triangle, then: $\mathbf{u} + \mathbf{v} = \mathbf{w}$
Vector Subtraction: $\overrightarrow{PQ} = (x_2 - x_1)\mathbf{i} + (y_2 - y_1)\mathbf{j} + (z_2 - z_1)\mathbf{k}$

Tips and Tricks

Finding Unit Vectors: Always calculate the magnitude first before dividing each component of the vector.
Direction Cosines Verification: Check if the sum of the squares of the direction cosines equals 1 to ensure correct angles.
Vector Addition: Visualize vectors geometrically to better understand their sum and how they relate to each other.
Practice Problems: Regularly solve problems involving vectors to strengthen understanding of concepts and their applications.

Summary

This exercise introduced several important concepts related to vectors, including their magnitude, unit vectors, and direction cosines. The exercise emphasized the use of formulas and verification methods for direction cosines, and highlighted how vector addition and subtraction can be applied in different contexts. The triangle law and vector operations are crucial tools for solving real-world physics and engineering problems involving vector quantities.

Reference

By Sir Shahzad Sair: