Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove the left distributive law, $$ \mathbf{u} \times(\mathbf{v}+\mathbf{w})=(\mathbf{u} \times \mathbf{v})+(\mathbf{u} \times \mathbf{w}) $$

Step-by-Step Solution

Answer

The left distributive law for vectors is proven as both sides equal.

Step 1: Define the Cross Product

The cross product of two vectors \(\mathbf{a} = (a_1, a_2, a_3)\) and \(\mathbf{b} = (b_1, b_2, b_3)\) is given by the determinant: \[\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} = (a_2b_3 - a_3b_2)\mathbf{i} - (a_1b_3 - a_3b_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}.\]

Step 2: Simplify \(\mathbf{v} + \mathbf{w}\)

Let \(\mathbf{v} = (v_1, v_2, v_3)\) and \(\mathbf{w} = (w_1, w_2, w_3)\). Then, the sum of these vectors is: \[\mathbf{v} + \mathbf{w} = (v_1 + w_1, v_2 + w_2, v_3 + w_3).\]

Step 3: Compute \(\mathbf{u} \times(\mathbf{v} + \mathbf{w})\)

Substitute \(\mathbf{v} + \mathbf{w}\) into the cross product formula: \[\mathbf{u} \times (\mathbf{v} + \mathbf{w}) = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ u_1 & u_2 & u_3 \ (v_1 + w_1) & (v_2 + w_2) & (v_3 + w_3) \end{vmatrix}.\] Simplify to get: \[(u_2(v_3 + w_3) - u_3(v_2 + w_2))\mathbf{i} - (u_1(v_3 + w_3) - u_3(v_1 + w_1))\mathbf{j} + (u_1(v_2 + w_2) - u_2(v_1 + w_1))\mathbf{k}.\]

Step 4: Expand Cross Product

Expand each term: - First component: \(u_2v_3 + u_2w_3 - u_3v_2 - u_3w_2\) - Second component: \(-(u_1v_3 + u_1w_3 - u_3v_1 - u_3w_1)\) - Third component: \(u_1v_2 + u_1w_2 - u_2v_1 - u_2w_1\).

Step 5: Compute \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{u} \times \mathbf{w}\) Separately

Compute individually: - \(\mathbf{u} \times \mathbf{v} = (u_2v_3 - u_3v_2)\mathbf{i} - (u_1v_3 - u_3v_1)\mathbf{j} + (u_1v_2 - u_2v_1)\mathbf{k}\); - \(\mathbf{u} \times \mathbf{w} = (u_2w_3 - u_3w_2)\mathbf{i} - (u_1w_3 - u_3w_1)\mathbf{j} + (u_1w_2 - u_2w_1)\mathbf{k}\).

Step 6: Calculate \((\mathbf{u} \times \mathbf{v}) + (\mathbf{u} \times \mathbf{w})\)

Add the two results from Step 5: - Combine the \(\mathbf{i}\)-components: \((u_2v_3 - u_3v_2) + (u_2w_3 - u_3w_2)\) - Combine the \(\mathbf{j}\)-components: \(-(u_1v_3 - u_3v_1) - (u_1w_3 - u_3w_1))\) - Combine the \(\mathbf{k}\)-components: \((u_1v_2 - u_2v_1) + (u_1w_2 - u_2w_1)\).

Step 7: Verify Identity

Compare the results of Steps 3 and 6. Both results yield the same vector components: \[u_2v_3 - u_3v_2 + u_2w_3 - u_3w_2,\ - (u_1v_3 - u_3v_1 + u_1w_3 - u_3w_1),\ u_1v_2 - u_2v_1 + u_1w_2 - u_2w_1\] This confirms the distributive property \(\mathbf{u} \times(\mathbf{v}+\mathbf{w})=(\mathbf{u} \times \mathbf{v})+(\mathbf{u} \times \\mathbf{w})\).

Key Concepts

Cross Product

The cross product is a type of vector multiplication that results in a vector, orthogonal to the two original input vectors. Imagine you have two vectors, such as 1. **Vector A: ** \(a_1, a_2, a_3\)2. **Vector B: ** \(b_1, b_2, b_3\)The cross product, \(\mathbf{a} \times \mathbf{b}\), can be determined using a special determinant formula. This formula involves organizing vector components into a matrix-like setup with unit vectors \(\mathbf{i}, \mathbf{j}, \mathbf{k}\). The result is another vector: - First component involves \(a_2b_3 - a_3b_2\) along \(\mathbf{i}\).- Second component involves \(-(a_1b_3 - a_3b_1)\) along \(\mathbf{j}\). - Third component involves \(a_1b_2 - a_2b_1\) along \(\mathbf{k}\).This operation is essential in physics and engineering as it finds applications in torque, rotational forces, and more.

Distributive Law

The distributive law is a fundamental principle in algebra and applies to the cross product of vectors. The left distributive law indicates that when you have a cross product, \(\mathbf{u} \times (\mathbf{v} + \mathbf{w})\), it distributes over addition: \((\mathbf{u} \times \mathbf{v}) + (\mathbf{u} \times \mathbf{w})\). This property simplifies complex problems by breaking them down into smaller, manageable parts. Think of it as expanding the cross product operation across a sum, ensuring you handle each part separately before polishing off the whole. Engineering and physics often rely on this property when dealing with vector fields.

Vector Addition

Vector addition is the operation of adding two vectors together to form a new vector, which is often represented in component form. To add two vectors, \(\mathbf{v} = (v_1, v_2, v_3)\) and \(\mathbf{w} = (w_1, w_2, w_3)\), you simply add each corresponding component:

• \(v_1 + w_1\) for the x-component
• \(v_2 + w_2\) for the y-component
• \(v_3 + w_3\) for the z-component

The resultant vector is \((v_1 + w_1, v_2 + w_2, v_3 + w_3)\).This process is intuitive, akin to adding numerical lists, and ensures that vector operations reflect real-world dynamics accurately.

Vector Algebra

Vector algebra is the branch of mathematics that focuses on vectors and their properties, including the operations that can be performed on them. This encompasses concepts like cross products, dot products, and vector addition. Vectors can be manipulated in a multidimensional space to model physical phenomena in engineering and physics. It's particularly useful: - For representing physical quantities like forces and velocities - In defining lines and planes in space - In transforming coordinate systems Understanding vector algebra equips one to solve complex problems, compute forces, or work with systems in motion. It's a crucial toolset in fields that require analytical thinking and application of mathematical principles to physical systems.