04_Ex 7.4

Exercise Questions

Questions	Links
Q1. Compute the cross product $\underline{a} \times \underline{b}$ and $\underline{b} \times \underline{a}$…	7.4 Q-1
Q2. Find $a$ unit vector perpendicular to the plane…	7.4 Q-2
Q3. Find the area of the triangle…	7.4 Q-3
Q4. Find the area of parallelogram…	7.4 Q-4
Q5. Which vectors are perpendicular or parallel…	7.4 Q-5
Q6. Prove that…	7.4 Q-6
Q7. If $\underline{a}+\underline{b}+\underline{c}=0$, then prove that…	7.4 Q-7
Q8. Prove that $\sin (\alpha-\beta)=\sin \alpha \cos \beta - \cos \alpha \sin \beta$	7.4 Q-8
Q9. If $\underline{a} \times \underline{b} = 0$ and $\mathbf{a} \cdot \mathbf{b} = 0$…	7.4 Q-9

Overview

In this exercise, we explore the properties and applications of the cross product (vector product) in vector algebra. The cross product is fundamental in understanding the relationship between vectors in three-dimensional space. It has various applications, including calculating areas of parallelograms and triangles, determining perpendicular vectors, and more.

Understanding the cross product is essential for working with forces in physics, computing the normal to a plane in geometry, and analyzing the rotation of objects in engineering.

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Key Concepts

1. Cross Product (Vector Product):
   The cross product of two vectors $\mathbf{u}$ and $\mathbf{v}$ is a vector perpendicular to both $\mathbf{u}$ and $\mathbf{v}$, with a magnitude proportional to the area of the parallelogram formed by the two vectors.

2. Properties of the Cross Product:

   • Non-Commutative: The order of the vectors in the cross product affects the direction.
   • Distributive: The cross product is distributive over vector addition.
   • Perpendicularity: The result of the cross product is perpendicular to the plane containing the original vectors.

3. Geometric Interpretation:
   The magnitude of the cross product is related to the area of the parallelogram formed by the vectors:

$$\text{Area} = |\mathbf{u} \times \mathbf{v}|$$

The direction of the cross product follows the right-hand rule.

4. Dot Product:
   The dot product (scalar product) is used to compute the angle between two vectors. It’s related to the cross product through the identity:

$$|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}| |\mathbf{v}| \sin \theta$$

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Important Formulas

1. Cross Product Formula:

$$\mathbf{u} \times \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \sin \theta , \mathbf{n}$$

where:

• $\mathbf{n}$ is the unit vector perpendicular to both $\mathbf{u}$ and $\mathbf{v}$,
• $\theta$ is the angle between $\mathbf{u}$ and $\mathbf{v}$.

2. Magnitude of the Cross Product:

$$|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}| |\mathbf{v}| \sin \theta$$

This represents the area of the parallelogram formed by $\mathbf{u}$ and $\mathbf{v}$.

3. Analytical Expression for Cross Product: For vectors in component form:

$$\mathbf{u} = u_1 \hat{i} + u_2 \hat{j} + u_3 \hat{k}, \quad \mathbf{v} = v_1 \hat{i} + v_2 \hat{j} + v_3 \hat{k}$$

The cross product is given by the determinant:

$$\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} u_1 & u_2 & u_3 v_1 & v_2 & v_3 \end{vmatrix}$$

Expanding gives:

$$\mathbf{u} \times \mathbf{v} = \left( (u_2 v_3 - u_3 v_2) \hat{i} - (u_1 v_3 - u_3 v_1) \hat{j} + (u_1 v_2 - u_2 v_1) \hat{k} \right)$$

4. Area of a Parallelogram:

$$\text{Area} = |\mathbf{u} \times \mathbf{v}|$$

5. Area of a Triangle:

$$\text{Area} = \frac{1}{2} |\mathbf{u} \times \mathbf{v}|$$

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Tips and Tricks

1. Visualizing the Cross Product:
   To understand the direction of the cross product, use the right-hand rule. Point your fingers in the direction of the first vector ($\mathbf{u}$), curl them toward the second vector ($\mathbf{v}$), and your thumb will point in the direction of the cross product.

2. Simplify the Determinant:
   When calculating the cross product using a determinant, ensure you expand correctly and track the signs carefully, as mistakes here can lead to incorrect results.

3. Perpendicular Vectors:
   If two vectors are perpendicular, their cross product will give the maximum possible magnitude ($|\mathbf{u}| |\mathbf{v}|$) since $\sin 90^\circ = 1$.

4. Unit Vector for Cross Product:
   The unit vector $\hat{n}$ in the cross product formula is determined using the right-hand rule. It is always perpendicular to the plane formed by the two vectors.

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Summary

This exercise focused on understanding and applying the cross product in vector algebra. The cross product is used to compute areas of parallelograms and triangles, and its properties such as non-commutativity, distributivity, and perpendicularity were discussed. The determinant formula for the cross product provides a straightforward method for computation in 3D space, and the right-hand rule helps determine its direction. Key applications of the cross product include determining perpendicular vectors, finding areas, and solving geometry problems involving vectors.

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Reference

By Sir Shahzad Sair: