Solving Vector Magnitude Expression

The problem asks us to evaluate the expression \({\left| {{\rm{\vec a}} \times {\rm{\vec b}}} \right|^2} + {\left| {{\rm{\vec a}} \cdot {\rm{\vec b}}} \right|^2}\), given the magnitudes of vectors \({\rm{\vec a}}}\) and \({\rm{\vec b}}}\). We are given that \({\left| {{\rm{\vec a}}} \right| = 2}\) and \({\left| {{\rm{\vec b}}} \right| = 3}\).

Understanding Vector Cross Product and Dot Product Magnitudes

Let \({\theta}\) be the angle between vectors \({\rm{\vec a}}}\) and \({\rm{\vec b}}}\). The magnitude of the cross product of two vectors \({\rm{\vec a}}}\) and \({\rm{\vec b}}}\) is defined as:

\({\left| {{\rm{\vec a}} \times {\rm{\vec b}}} \right| = \left| {{\rm{\vec a}}}} \right|\left| {{\rm{\vec b}}}} \right|\sin \theta\)

The dot product of two vectors \({\rm{\vec a}}}\) and \({\rm{\vec b}}}\) is defined as:

\({{\rm{\vec a}} \cdot {\rm{\vec b}}} = \left| {{\rm{\vec a}}}} \right|\left| {{\rm{\vec b}}}} \right|\cos \theta\)

Evaluating the Expression

We need to find \({\left| {{\rm{\vec a}} \times {\rm{\vec b}}} \right|^2} + {\left| {{\rm{\vec a}} \cdot {\rm{\vec b}}} \right|^2}\). Let's square the magnitudes of the cross product and the dot product:

\({\left| {{\rm{\vec a}} \times {\rm{\vec b}}} \right|^2} = {\left( {\left| {{\rm{\vec a}}}} \right|\left| {{\rm{\vec b}}}} \right|\sin \theta } \right)^2} = {\left| {{\rm{\vec a}}}} \right|^2}{\left| {{\rm{\vec b}}}} \right|^2}{\sin ^2}\theta\)

\({\left| {{\rm{\vec a}} \cdot {\rm{\vec b}}} \right|^2} = {\left( {\left| {{\rm{\vec a}}}} \right|\left| {{\rm{\vec b}}}} \right|\cos \theta } \right)^2} = {\left| {{\rm{\vec a}}}} \right|^2}{\left| {{\rm{\vec b}}}} \right|^2}{\cos ^2}\theta\)

Now, let's add these two squared terms:

\({\left| {{\rm{\vec a}} \times {\rm{\vec b}}} \right|^2} + {\left| {{\rm{\vec a}} \cdot {\rm{\vec b}}} \right|^2} = {\left| {{\rm{\vec a}}}} \right|^2}{\left| {{\rm{\vec b}}}} \right|^2}{\sin ^2}\theta + {\left| {{\rm{\vec a}}}} \right|^2}{\left| {{\rm{\vec b}}}} \right|^2}{\cos ^2}\theta\)

We can factor out the common term \({\left| {{\rm{\vec a}}}} \right|^2}{\left| {{\rm{\vec b}}}} \right|^2}\):

\({\left| {{\rm{\vec a}} \times {\rm{\vec b}}} \right|^2} + {\left| {{\rm{\vec a}} \cdot {\rm{\vec b}}} \right|^2} = {\left| {{\rm{\vec a}}}} \right|^2}{\left| {{\rm{\vec b}}}} \right|^2}\left( {{{\sin }^2}\theta + {{\cos }^2}\theta } \right)\)

Using the fundamental trigonometric identity \({\sin ^2}\theta + {\cos ^2}\theta = 1\), the expression simplifies to:

\({\left| {{\rm{\vec a}} \times {\rm{\vec b}}} \right|^2} + {\left| {{\rm{\vec a}} \cdot {\rm{\vec b}}} \right|^2} = {\left| {{\rm{\vec a}}}} \right|^2}{\left| {{\rm{\vec b}}}} \right|^2}\left( 1 \right) = {\left| {{\rm{\vec a}}}} \right|^2}{\left| {{\rm{\vec b}}}} \right|^2}\)

Substituting Given Magnitudes

We are given \({\left| {{\rm{\vec a}}} \right| = 2}\) and \({\left| {{\rm{\vec b}}} \right| = 3}\).

So, \({\left| {{\rm{\vec a}}}} \right|^2 = {2^2} = 4}\) and \({\left| {{\rm{\vec b}}}} \right|^2 = {3^2} = 9}\).

Substituting these values into the simplified expression:

\({\left| {{\rm{\vec a}} \times {\rm{\vec b}}} \right|^2} + {\left| {{\rm{\vec a}} \cdot {\rm{\vec b}}} \right|^2} = {\left| {{\rm{\vec a}}}} \right|^2}{\left| {{\rm{\vec b}}}} \right|^2} = 4 \times 9 = 36\)

Thus, the value of the expression \({\left| {{\rm{\vec a}} \times {\rm{\vec b}}} \right|^2} + {\left| {{\rm{\vec a}} \cdot {\rm{\vec b}}} \right|^2}\) is 36.

Summary of Steps

1. Recall the definitions of \({\left| {{\rm{\vec a}} \times {\rm{\vec b}}} \right|}\) and \({\vec a} \cdot {\vec b}}\) in terms of magnitudes and the angle between the vectors.
2. Square both expressions.
3. Add the squared expressions.
4. Factor out common terms.
5. Use the trigonometric identity \({\sin ^2}\theta + {\cos ^2}\theta = 1\).
6. Substitute the given magnitudes of the vectors.
7. Calculate the final result.

Final Result

The calculated value of \({\left| {{\rm{\vec a}} \times {\rm{\vec b}}} \right|^2} + {\left| {{\rm{\vec a}} \cdot {\rm{\vec b}}} \right|^2}\) is 36.

Revision Table: Vector Product Properties

Additional Information: Vector Identities and Applications

The identity \({\left| {{\rm{\vec a}} \times {\rm{\vec b}}} \right|^2} + {\left| {{\rm{\vec a}} \cdot {\rm{\vec b}}} \right|^2} = {\left| {{\rm{\vec a}}}} \right|^2}{\left| {{\rm{\vec b}}}} \right|^2}\) is a useful vector identity derived directly from the definitions of the dot and cross products and the trigonometric identity \({\sin ^2}\theta + {\cos ^2}\theta = 1\). This identity holds true for any two vectors \({\vec a}\) and \({\vec b}\) in 3D space.

Understanding the relationship between the dot product, cross product, and vector magnitudes is crucial in various physics and engineering applications, such as calculating work done by a force (using dot product) or calculating torque (using cross product).