Problem 28 - Advanced Engineering Mathematics

Question

Find the indicated scalar or vector without using \((5),(13)\), or \((15)\). $$ \mathbf{i} \cdot[\mathbf{j} \times(-\mathbf{k})] $$

Step-by-Step Solution

Answer

-1

Step 1: Understand the Problem

We need to find the result of the vector expression \( \mathbf{i} \cdot [ \mathbf{j} \times (-\mathbf{k}) ] \). This involves the dot product of \( \mathbf{i} \) and the cross product of \( \mathbf{j} \) and \( -\mathbf{k} \).

Step 2: Compute the Inner Cross Product

Compute \( \mathbf{j} \times (-\mathbf{k}) \). Using the right-hand rule or the determinant method: \[ \mathbf{j} \times (-\mathbf{k}) = -[\mathbf{j} \times \mathbf{k}] = -\mathbf{i} \].

Step 3: Resolve the Outer Dot Product

Now compute \( \mathbf{i} \cdot (-\mathbf{i}) \). The dot product of a vector with its negative is the negative of the square of its magnitude: \[ \mathbf{i} \cdot (-\mathbf{i}) = -1 \], as the magnitude of \( \mathbf{i} \) is 1.

Step 4: Conclude the Solution

From the calculations, we have determined that \( \mathbf{i} \cdot [\mathbf{j} \times (-\mathbf{k})] = -1 \).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation is useful in many areas of physics and mathematics, especially in vector calculus.
The formula for the dot product of two vectors \ \( \mathbf{a} = (a_1, a_2, a_3) \ \) and \ \( \mathbf{b} = (b_1, b_2, b_3) \ \) is given by:
\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]
In simpler terms, it involves multiplying the corresponding components of two vectors and summing the results. This calculation provides a measure of the extent to which the vectors align.

• If the dot product is positive, the vectors are pointing in generally the same direction.
• If it's negative, they point in opposite directions.
• If it is zero, the vectors are perpendicular.

The dot product is crucial because it allows us to calculate quantities such as the length of a vector and the angle between two vectors.

Cross Product

The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space. Unlike the dot product, the cross product results in a vector rather than a scalar.
The formula for the cross product of two vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \) is:
\[ \mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2)\mathbf{i} + (a_3b_1 - a_1b_3)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k} \]
This formula generates a new vector that is perpendicular to the plane containing \( \mathbf{a} \) and \( \mathbf{b} \). The length of this vector corresponds to the area of the parallelogram spanned by the original vectors. Some key points about the cross product include:

• The cross product is zero if and only if the vectors are parallel.
• It is anti-commutative, meaning \( \mathbf{a} \times \mathbf{b} = - (\mathbf{b} \times \mathbf{a}) \).
• The direction of the resulting vector is determined using the Right-Hand Rule.

The cross product finds applications in calculating torque, rotating vectors, and in computer graphics for rendering three-dimensional objects.

Right-Hand Rule

The Right-Hand Rule is a simple mnemonic for determining the direction of the vector resulting from the cross product. Understanding the right-hand rule is essential in vector calculus where orientation in three-dimensional space is crucial.
To apply the Right-Hand Rule, follow these steps:

• Point your index finger in the direction of the first vector \( \mathbf{a} \).
• Point your middle finger in the direction of the second vector \( \mathbf{b} \).
• Your thumb will then point in the direction of the resulting cross product \( \mathbf{a} \times \mathbf{b} \).

It's important to note that your fingers must follow the sequence correctly. This technique ensures that the resultant vector is perpendicular to both input vectors.
This rule is pivotal when working with electromagnetism, predicting the behavior of rotational systems, and understanding various physical phenomena involving vector orientation.