Question

Prove in two ways that for scalars \(a\) and \(b\) \((a \mathbf{u}) \times(b \mathbf{v})=a b(\mathbf{u} \times \mathbf{v}) .\) Use the definition of the cross product and the determinant formula.

Short answer

Question: Prove that for any scalar values a and b, and vectors u and v, the equation (a u) x (b v) = a b (u x v) is true. Short Answer: Using the definition of the cross product and the determinant formula, we can expand and simplify both sides of the equation to show that (a u) x (b v) = a b (u x v) is indeed true.

Step-by-step solution

Method 1: Definition of the cross product

Let \(\mathbf{u} = u_1 \mathbf{i} + u_2 \mathbf{j} + u_3 \mathbf{k}\) and \(\mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k}\). Also, let \(\mathbf{w} = a\mathbf{u}\) and \(\mathbf{x} = b\mathbf{v}\), so \(\mathbf{w} \times \mathbf{x}\) is equivalent to \((a \mathbf{u}) \times(b \mathbf{v})\). Step 1: Calculate \(\mathbf{w} = a\mathbf{u}\) and \(\mathbf{x} = b\mathbf{v}\) $$\mathbf{w} = a(u_1 \mathbf{i} + u_2 \mathbf{j} + u_3 \mathbf{k}) = a u_1 \mathbf{i} + a u_2 \mathbf{j} + a u_3 \mathbf{k}$$ $$\mathbf{x} = b(v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k}) = b v_1 \mathbf{i} + b v_2 \mathbf{j} + b v_3 \mathbf{k}$$ Step 2: Calculate the cross product \(\mathbf{w} \times \mathbf{x}\) $$\mathbf{w} \times \mathbf{x} = \begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a u_1 & a u_2 & a u_3 \\ b v_1 & b v_2 & b v_3 \end{bmatrix} = a b \begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{bmatrix} = ab(\mathbf{u} \times \mathbf{v})$$ Thus, \((a \mathbf{u}) \times(b \mathbf{v}) = a b(\mathbf{u} \times \mathbf{v})\).

Method 2: Using the Determinant Formula

Step 1: Calculate the determinant of a 3x3 matrix for \(\mathbf{w} \times \mathbf{x}\) We have \(\mathbf{w} = a u_1 \mathbf{i} + a u_2 \mathbf{j} + a u_3 \mathbf{k}\) and \(\mathbf{x} = b v_1 \mathbf{i} + b v_2 \mathbf{j} + b v_3 \mathbf{k}\). $$\mathbf{w} \times \mathbf{x} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a u_1 & a u_2 & a u_3 \\ b v_1 & b v_2 & b v_3 \end{vmatrix} = a b \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix} = ab(\mathbf{u} \times \mathbf{v})$$ Thus, \((a \mathbf{u}) \times(b \mathbf{v}) = a b(\mathbf{u} \times \mathbf{v})\).

Key concepts

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a single number). This operation scales the vector, effectively stretching or compressing it without changing its direction.
For a vector \( \mathbf{u} = u_1 \mathbf{i} + u_2 \mathbf{j} + u_3 \mathbf{k} \) and a scalar \( a \), the result of scalar multiplication is \( a\mathbf{u} = a u_1 \mathbf{i} + a u_2 \mathbf{j} + a u_3 \mathbf{k} \).
This means each component of the vector is multiplied by the scalar \( a \). Hence, the resulting vector maintains the same direction but its magnitude is scaled by \( a \).

• If \( a > 1 \), the vector is stretched.
• If \( 0 < a < 1 \), the vector is compressed.
• If \( a < 0 \), the vector is reversed in direction.

As shown in the exercise, scalar multiplication prepares the vectors for further operations such as cross products.

Vector Operations

Vector operations, specifically the cross product, are fundamental in vector calculus. The cross product \( \mathbf{u} \times \mathbf{v} \) yields a vector that is perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \).
This operation is defined in terms of the determinants of a matrix representing the vectors involved. Given two vectors \( \mathbf{u} = u_1 \mathbf{i} + u_2 \mathbf{j} + u_3 \mathbf{k} \) and \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k} \), their cross product is calculated as follows:
\[ \mathbf{u} \times \mathbf{v} = \begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \end{bmatrix} \]
The resulting vector is found using the determinant method. This operation is useful in physics and engineering to find a vector that is orthogonal to the plane formed by two vectors.

Determinant Formula

In vector calculus, the determinant formula plays a vital role in finding the cross product of vectors. The cross product can be expressed using a 3x3 matrix determinant containing unit vectors and the components of the vectors involved.
For vectors \( \mathbf{u} \) and \( \mathbf{v} \), the formula is written as:
\[ \mathbf{w} \times \mathbf{x} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \end{vmatrix} \]
To calculate this determinant, expand along the top row (containing the unit vectors):

• First, isolate \( \mathbf{i} \) and find the determinant of the remaining 2x2 matrix.
• Then, subtract \( \mathbf{j} \) times the determinant of its corresponding 2x2 matrix.
• Finally, add \( \mathbf{k} \) multiplied by its associated determinant.

These computations give the components of the new vector orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \). Calculating cross products this way is crucial for verifying vector properties such as orthogonality.

Vector Calculus

Vector calculus studies how vector fields change over space and is crucial in physics and engineering. It involves differentiating and integrating vector fields to find various physical quantities. The cross product is among the essential vector operations in this field.
In our exercise, we explored how scalar multiplication and cross product relate closely in vector calculus. The property \( (a \mathbf{u}) \times (b \mathbf{v}) = ab(\mathbf{u} \times \mathbf{v}) \) exemplifies this by showing how scaling vectors by scalars before taking their cross product scales the result by the product of these scalars.
Understanding such principles aids in fields like electromagnetism where vector calculus is used extensively. Analyzing vector fields helps describe distributions of force, velocity, and other vector quantities across regions in space.