Question

State whether or not \(\mathbf{A}=\mathbf{B}\) and, if not, what conditions are imposed on \(\mathbf{A}\) and \(\mathbf{B}\) when \((a) \mathbf{A} \cdot \mathbf{a}_{x}=\mathbf{B} \cdot \mathbf{a}_{x} ;(b) \mathbf{A} \times \mathbf{a}_{x}=\mathbf{B} \times \mathbf{a}_{x} ;(c) \mathbf{A} \cdot \mathbf{a}_{x}=\mathbf{B} \cdot \mathbf{a}_{x}\) and \(\mathbf{A} \times \mathbf{a}_{x}=\mathbf{B} \times \mathbf{a}_{x} ;(d) \mathbf{A} \cdot \mathbf{C}=\mathbf{B} \cdot \mathbf{C}\) and \(\mathbf{A} \times \mathbf{C}=\mathbf{B} \times \mathbf{C}\) where \(\mathbf{C}\) is any vector except \(\mathbf{C}=0\).

Short answer

Answer: We can conclude that \(\mathbf{A}=\mathbf{B}\) if \(\mathbf{A} \cdot \mathbf{C} = \mathbf{B} \cdot \mathbf{C}\) and \(\mathbf{A} \times \mathbf{C} = \mathbf{B} \times \mathbf{C}\) for any vector \(\mathbf{C} \neq 0\).

Step-by-step solution

Part (a): \(\mathbf{A} \cdot \mathbf{a}_{x} = \mathbf{B} \cdot \mathbf{a}_{x}\)

For two vectors to have the same dot product with a specific vector \(\mathbf{a}_{x}\), they must have the same component along the direction of \(\mathbf{a}_{x}\). However, this alone is not sufficient to determine that \(\mathbf{A}=\mathbf{B}\). Their other components, along different directions, could still be different. Therefore, we cannot conclude that \(\mathbf{A}=\mathbf{B}\) in this case.

Part (b): \(\mathbf{A} \times \mathbf{a}_{x} = \mathbf{B} \times \mathbf{a}_{x}\)

For two vectors to have the same cross product with a specific vector \(\mathbf{a}_{x}\), they must be either parallel or antiparallel to each other, or both be zero vectors. However, this still does not guarantee that \(\mathbf{A} = \mathbf{B}\). They can have different magnitudes or be pointing in opposite directions, as long as they remain parallel or antiparallel.

Part (c): \(\mathbf{A} \cdot \mathbf{a}_{x} = \mathbf{B} \cdot \mathbf{a}_{x}\) and \(\mathbf{A} \times \mathbf{a}_{x} = \mathbf{B} \times \mathbf{a}_{x}\)

In this case, we have both the conditions from parts (a) and (b). This means \(\mathbf{A}\) and \(\mathbf{B}\) have the same component along \(\mathbf{a}_{x}\) and are parallel or antiparallel. However, this is still not sufficient to conclude that \(\mathbf{A}=\mathbf{B}\). They can still have different magnitudes or be pointing in opposite directions.

Part (d): \(\mathbf{A} \cdot \mathbf{C} = \mathbf{B} \cdot \mathbf{C}\) and \(\mathbf{A} \times \mathbf{C} = \mathbf{B} \times \mathbf{C}\) for any vector \(\mathbf{C} \neq 0\)

Since these conditions apply for any vector \(\mathbf{C}\) that is not a zero vector, this means that \(\mathbf{A}\) and \(\mathbf{B}\) have the same dot and cross products with all possible vectors. This means, geometrically, that they have the same angle with any other vector and the same magnitude of the cross products, which can only happen if \(\mathbf{A}=\mathbf{B}\). So, in this case, we can conclude that \(\mathbf{A}=\mathbf{B}\).

Key concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It is used to determine the angle between two vectors or to project one vector onto another. The formula for the dot product of vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by \( \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta \), where \( \theta \) is the angle between the two vectors. The result of a dot product is a scalar, not a vector.

When vectors have the same dot product with a specific vector, it indicates that they have an equal component along the direction of that vector. However, this condition alone doesn't guarantee that the vectors are equal across all dimensions. They might still differ in components not aligned to the specific reference vector.

Understanding the dot product is crucial for tasks such as assessing vector angles and determining vector component alignment.

Cross Product

The cross product, also called the vector product, is another essential vector operation. It applies only in three-dimensional space and results in a vector. Unlike the dot product, the cross product will give you a vector that is perpendicular to both original vectors. The formula is \( \mathbf{A} \times \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \sin \theta \ \mathbf{n} \), where \( \mathbf{n} \) is the unit vector perpendicular to the plane containing \( \mathbf{A} \) and \( \mathbf{B} \), and \( \theta \) is the angle between them.

Having the same cross product with a specific vector indicates that the vectors are parallel, antiparallel, or possibly zero vectors. This means that they lie along the same line of action and may have different magnitudes or orientations. You're not assured of their equality unless more conditions are considered.

Cross products are particularly useful in physics for torque calculations and determining perpendicular directions.

Vector Magnitude

The magnitude of a vector is a measure of its length or size. For a vector \( \mathbf{A} = \langle a_1, a_2, a_3 \rangle \), its magnitude is calculated using the formula \( |\mathbf{A}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \).

Magnitude is a key attribute of vectors, different from direction, and is crucial in determining how much of a certain entity (force, velocity, etc.) the vector represents. Even if two vectors have the same direction, they can still have different magnitudes, leading to different vector qualities.

When solving problems with vectors, always consider both their magnitude and direction to determine their true equivalence or nature. This distinction is particularly vital when dealing with vector addition, subtraction, and multiplication.

Parallel Vectors

Parallel vectors have the same or exact opposite direction. Two vectors \( \mathbf{A} \) and \( \mathbf{B} \) are parallel if there exists a scalar \( k \) such that \( \mathbf{A} = k\mathbf{B} \). Parallel vectors can have different magnitudes, but visually they stretch along the same line or direction.

When dealing with cross products, parallel vectors result in a zero vector since the sine of their included angle is zero (\( \sin 0 = 0 \)). Recognizing when vectors are parallel is essential in physics and engineering, where alignment and orientation play a critical role.

Keep in mind that parallel vectors can still differ from each other by their lengths or by pointing in exactly opposite directions. Therefore, identifying vectors as parallel only suggests a specific directional alignment rather than full vector equality.