Problem 56 - Advanced Engineering Mathematics

Question

Two vectors a and b lie in the \(x z\) -plane so that the angle between them is \(120^{\circ} .\) If \(\|\mathbf{a}\|=\sqrt{27}\) and \(\|\mathbf{b}\|=8\), find all possible values of \(\mathbf{a} \times \mathbf{b}\).

Step-by-Step Solution

Answer

The possible values of \( \mathbf{a} \times \mathbf{b} \) are \( 12\sqrt{3}\mathbf{j} \) and \( -12\sqrt{3}\mathbf{j} \).

Step 1: Identify the Formula for Cross Product Magnitude

The magnitude of the cross product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by \( \| \mathbf{a} \times \mathbf{b} \| = \| \mathbf{a} \| \| \mathbf{b} \| \sin \theta \), where \( \theta \) is the angle between the vectors.

Step 2: Substitute Given Values into the Formula

Given that \( \| \mathbf{a} \| = \sqrt{27} \), \( \| \mathbf{b} \| = 8 \), and \( \theta = 120^{\circ} \), substitute these into the formula: \[\| \mathbf{a} \times \mathbf{b} \| = (\sqrt{27})(8)\sin(120^{\circ})\] Since \( \sin(120^{\circ}) = \frac{\sqrt{3}}{2} \), the expression becomes:\[\| \mathbf{a} \times \mathbf{b} \| = \sqrt{27} \cdot 8 \cdot \frac{\sqrt{3}}{2}\]

Step 3: Calculate the Magnitude

To find the magnitude of the cross product, simplify the expression:\[\| \mathbf{a} \times \mathbf{b} \| = \frac{\sqrt{27} \cdot 8 \cdot \sqrt{3}}{2} = \frac{8 \cdot 3 \cdot \sqrt{3}}{2} = 12 \cdot \sqrt{3} = 12 \sqrt{3}\] Thus, the magnitude of \( \mathbf{a} \times \mathbf{b} \) is \( 12 \sqrt{3} \).

Step 4: Determine Possible Directions

Since \( \mathbf{a} \) and \( \mathbf{b} \) lie in the \( xz \)-plane, \( \mathbf{a} \times \mathbf{b} \) will be perpendicular to that plane, meaning it will be along the positive or negative \( y \)-axis. Thus, \( \mathbf{a} \times \mathbf{b} \) could either be \( 12 \sqrt{3} \mathbf{j} \) or \( -12 \sqrt{3} \mathbf{j} \).

Key Concepts

Angle Between Vectors

When we talk about the angle between two vectors, we're referring to the geometric opening between these two lines originating from a common point. This angle is often denoted by \( \theta \), which stands for the Greek letter Theta. In problems centered on vector cross products, this angle plays a crucial role.

For two vectors \( \mathbf{a} \) and \( \mathbf{b} \), the angle between them can determine the directional behavior and magnitude of their resultant cross product. When given the angle \( 120^\circ \), as in our example, it falls into the category of obtuse angles (angles greater than \( 90^\circ \) but less than \( 180^\circ \)).

The sine function (\( \sin \theta \)) is intrinsic to calculating cross product magnitudes. It's worth remembering:

• \( \sin(90^\circ) = 1 \),
• \( \sin(120^\circ) = \frac{\sqrt{3}}{2} \), which was specifically used in our calculation.

These sine values help us understand how much of each vector 'contributes' to their cross product, particularly when they aren't perpendicular.

Lastly, knowing this angle aids us in evaluating how much "twist" or rotational difference exists between vectors.

Magnitude of Vectors

To discuss the magnitude of vectors, imagine each vector as an arrow. The magnitude resembles the length of this arrow, indicating the vector's strength or size.
The magnitude of a vector is typically represented by two vertical bars around the vector, like this: \( \| \mathbf{a} \| \). In the problem, it's given as \( \sqrt{27} \) for vector \( \mathbf{a} \) and as \( 8 \) for vector \( \mathbf{b} \).

In terms of calculations, the magnitude influences how large or small the product of two vectors might be. When computing the magnitude of their cross product:

• The magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \) are multiplied together,
• Then multiplied by \( \sin \theta \), the sine of the angle between them (\( 120^\circ \)).

This calculation unfolds into:\[\| \mathbf{a} \times \mathbf{b} \| = \| \mathbf{a} \| \| \mathbf{b} \| \sin \theta\] which simplifies to \( 12 \sqrt{3} \) in our exercise.

Understanding magnitudes allows us to interpret and predict the strength of vector interactions, whether in physics, engineering, or geometry.

Direction of Cross Product

The direction of the cross product is a fundamental attribute that sets it apart from scalar multiplication. While magnitudes give us size, direction provides orientation in three-dimensional space.

By definition, the cross product \( \mathbf{a} \times \mathbf{b} \) results in a vector perpendicular to the plane containing \( \mathbf{a} \) and \( \mathbf{b} \). In our case, since both \( \mathbf{a} \) and \( \mathbf{b} \) lie on the \( xz \) plane, the cross product must point directly out of or into this plane. This leads us to the positive or negative \( y \)-axis.

Here's what this looks like in terms of vectors:

• If \( \mathbf{a} \times \mathbf{b} = 12\sqrt{3} \mathbf{j} \), it points along the positive \( y \)-direction.
• If \( \mathbf{a} \times \mathbf{b} = -12\sqrt{3} \mathbf{j} \), it points along the negative \( y \)-direction.

Using the right-hand rule is a practical method to visualize this direction:

• Point your index finger in the direction of \( \mathbf{a} \),
• And your middle finger in the direction of \( \mathbf{b} \),
• Your thumb then points in the direction of \( \mathbf{a} \times \mathbf{b} \).

This principle clarifies how vector directions work in cross products, providing students with a physical grasp of three-dimensional vector dynamics.