Understanding Vector Equations and Properties

This question asks us to evaluate the correctness of three given vector equations involving vector addition, subtraction, dot product, cross product, and vector magnitudes. Let's analyze each statement carefully using fundamental vector properties.

Analyzing Statement 1: Dot Product of Sum and Difference

Statement 1 is: \( \left( \vec{a}+\vec{b} \right)\cdot \left( \vec{a}-\vec{b} \right)={{\left| {\vec{a}} \right|}^{2}}-{{\left| {\vec{b}} \right|}^{2}} \)

To verify this, we can expand the left side using the distributive property of the dot product, similar to how we expand algebraic expressions:

Left Side (LS) \( = \left( \vec{a}+\vec{b} \right)\cdot \left( \vec{a}-\vec{b} \right) \)

\( = \vec{a} \cdot \vec{a} + \vec{a} \cdot (-\vec{b}) + \vec{b} \cdot \vec{a} + \vec{b} \cdot (-\vec{b}) \)

\( = \vec{a} \cdot \vec{a} - \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} - \vec{b} \cdot \vec{b} \)

We know that \( \vec{a} \cdot \vec{a} = |\vec{a}|^2 \) and \( \vec{b} \cdot \vec{b} = |\vec{b}|^2 \). Also, the dot product is commutative, meaning \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \). Substituting these into the expression:

\( = |\vec{a}|^2 - \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{b} - |\vec{b}|^2 \)

\( = |\vec{a}|^2 - |\vec{b}|^2 \)

This matches the Right Side (RS) of the equation. Therefore, statement 1 is correct.

Analyzing Statement 2: Product of Magnitudes of Sum and Difference

Statement 2 is: \( \left( \left| \vec{a}+\vec{b} \right| \right)\left( \left| \vec{a}-\vec{b} \right| \right)={{\left| {\vec{a}} \right|}^{2}}-{{\left| {\vec{b}} \right|}^{2}} \)

Let's consider the squares of the magnitudes first:

\( |\vec{a}+\vec{b}|^2 = (\vec{a}+\vec{b}) \cdot (\vec{a}+\vec{b}) = \vec{a} \cdot \vec{a} + \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{b} = |\vec{a}|^2 + 2(\vec{a} \cdot \vec{b}) + |\vec{b}|^2 \)

\( |\vec{a}-\vec{b}|^2 = (\vec{a}-\vec{b}) \cdot (\vec{a}-\vec{b}) = \vec{a} \cdot \vec{a} - \vec{a} \cdot \vec{b} - \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{b} = |\vec{a}|^2 - 2(\vec{a} \cdot \vec{b}) + |\vec{b}|^2 \)

The left side of statement 2 is \( \left| \vec{a}+\vec{b} \right| \left| \vec{a}-\vec{b} \right| \). If we square this, we get \( \left( \left| \vec{a}+\vec{b} \right| \left| \vec{a}-\vec{b} \right| \right)^2 = |\vec{a}+\vec{b}|^2 |\vec{a}-\vec{b}|^2 \). For statement 2 to be correct, its square must equal the square of the right side:

\( (|\vec{a}|^2 - |\vec{b}|^2)^2 = (|\vec{a}|^2)^2 - 2|\vec{a}|^2 |\vec{b}|^2 + (|\vec{b}|^2)^2 = |\vec{a}|^4 - 2|\vec{a}|^2 |\vec{b}|^2 + |\vec{b}|^4 \)

Now let's look at \( |\vec{a}+\vec{b}|^2 |\vec{a}-\vec{b}|^2 \):

\( = (|\vec{a}|^2 + 2(\vec{a} \cdot \vec{b}) + |\vec{b}|^2) (|\vec{a}|^2 - 2(\vec{a} \cdot \vec{b}) + |\vec{b}|^2) \)

Let \( X = |\vec{a}|^2 + |\vec{b}|^2 \) and \( Y = 2(\vec{a} \cdot \vec{b}) \). The expression becomes \( (X+Y)(X-Y) = X^2 - Y^2 \).

\( = (|\vec{a}|^2 + |\vec{b}|^2)^2 - (2(\vec{a} \cdot \vec{b}))^2 \)

\( = (|\vec{a}|^4 + 2|\vec{a}|^2 |\vec{b}|^2 + |\vec{b}|^4) - 4(\vec{a} \cdot \vec{b})^2 \)

Comparing this with \( |\vec{a}|^4 - 2|\vec{a}|^2 |\vec{b}|^2 + |\vec{b}|^4 \), they are clearly not equal for general vectors unless \( 4|\vec{a}|^2 |\vec{b}|^2 - 4(\vec{a} \cdot \vec{b})^2 = 0 \), which is not always true. For instance, if \(\vec{a}\) and \(\vec{b}\) are orthogonal (\(\vec{a} \cdot \vec{b} = 0\)), then \( |\vec{a}+\vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 \) and \( |\vec{a}-\vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 \). The left side of statement 2 becomes \( \sqrt{|\vec{a}|^2 + |\vec{b}|^2} \sqrt{|\vec{a}|^2 + |\vec{b}|^2} = |\vec{a}|^2 + |\vec{b}|^2 \). The right side is \( |\vec{a}|^2 - |\vec{b}|^2 \). These are only equal if \( 2|\vec{b}|^2 = 0 \), i.e., \( |\vec{b}|=0 \). Thus, statement 2 is not correct for all vectors \(\vec{a}\) and \(\vec{b}\).

Analyzing Statement 3: Relation between Dot Product and Cross Product Magnitudes

Statement 3 is: \( {{\left| \vec{a}\cdot \vec{b} \right|}^{2}}+{{\left| \vec{a}\times \vec{b} \right|}^{2}}={{\left| {\vec{a}} \right|}^{2}}{{\left| {\vec{b}} \right|}^{2}} \)

We know the definitions of the dot product and the magnitude of the cross product in terms of the angle \( \theta \) between vectors \(\vec{a}\) and \(\vec{b}\):

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

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

Substitute these into the left side of statement 3:

LS \( = |\vec{a} \cdot \vec{b}|^2 + |\vec{a} \times \vec{b}|^2 \)

Since the terms inside the magnitudes are already magnitudes, the absolute value bars around the dot product and cross product magnitude are redundant when squaring, but let's keep them for consistency with the statement notation:

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

Assuming \( \theta \) is in the range \( 0 \le \theta \le \pi \), \( \sin \theta \ge 0 \). \( \cos \theta \) can be positive or negative. However, squaring makes the result positive.

\( = (|\vec{a}| |\vec{b}| \cos \theta)^2 + (|\vec{a}| |\vec{b}| \sin \theta)^2 \)

\( = |\vec{a}|^2 |\vec{b}|^2 \cos^2 \theta + |\vec{a}|^2 |\vec{b}|^2 \sin^2 \theta \)

\( = |\vec{a}|^2 |\vec{b}|^2 (\cos^2 \theta + \sin^2 \theta) \)

Using the trigonometric identity \( \cos^2 \theta + \sin^2 \theta = 1 \):

\( = |\vec{a}|^2 |\vec{b}|^2 (1) \)

\( = |\vec{a}|^2 |\vec{b}|^2 \)

This matches the Right Side (RS) of the equation. Therefore, statement 3 is correct. This identity is sometimes known as Lagrange's identity in vector form.

Summary of Verification

Based on our analysis, statements 1 and 3 are correct, while statement 2 is incorrect.

Revision Table: Key Vector Properties

Additional Information: Understanding Vector Operations

Vector operations like the dot product and cross product are fundamental in physics and engineering. They provide ways to multiply vectors, resulting in either a scalar (dot product) or a vector (cross product).

• Dot Product: The dot product of two vectors is a scalar quantity. It is related to the projection of one vector onto another and is used to calculate work done by a force or the power of a force. Geometrically, it involves the cosine of the angle between the vectors.
• Cross Product: The cross product of two vectors is a vector quantity that is perpendicular to both original vectors. Its magnitude is related to the area of the parallelogram formed by the vectors. It is used in calculations involving torque, angular momentum, and magnetic force. Geometrically, it involves the sine of the angle between the vectors.
• Magnitude: The magnitude of a vector represents its length or size. It is a scalar quantity calculated using the Pythagorean theorem in component form, or through the dot product of the vector with itself \( (|\vec{v}| = \sqrt{\vec{v} \cdot \vec{v}}) \).

Understanding these operations and their properties, such as the distributive property for dot product or the relationship between dot and cross product magnitudes shown in statement 3, is crucial for solving problems in vector algebra.