Question

If \(\vec{a}, \vec{b}, \vec{c}\) and \(\vec{d}\) are unit vectors such that $$ (\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})=1 $$ and $$ \vec{a} \cdot \vec{c}=\frac{1}{2}, $$ then (A) \(\vec{a}, \vec{b}, \vec{c}\) are non-coplanar (B) \(\vec{b}, \vec{c}, \vec{d}\) are non-coplanar (C) \(\vec{b}, d\) are non-parallel (D) \(\vec{a}, \vec{d}\) are parallel and \(b, \vec{c}\) are paraliel

Short answer

(A) Correct, (B) Correct, (C) Correct, (D) Incorrect; there is no evidence that \(\vec{a}, \vec{d}\) are parallel, nor is there evidence that \(\vec{b}, \vec{c}\) are parallel.

Step-by-step solution

Understand the Cross Product

The cross product of two vectors results in a vector that is perpendicular to the plane formed by the original vectors. If \(\vec{a} \times \vec{b}\) is non-zero, it means that \(\vec{a}\) and \(\vec{b}\) are non-parallel and form a plane. Similarly, if \(\vec{c} \times \vec{d}\) is non-zero, then \(\vec{c}\) and \(\vec{d}\) form another plane.

Dot Product of Cross Products

Given the equation \( (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = 1 \), this implies that the cross products \(\vec{a} \times \vec{b}\) and \(\vec{c} \times \vec{d}\) are non-zero vectors. Additionally, since their dot product is 1 (a positive non-zero scalar), the vectors are non-orthogonal and have some component along the same direction.

Analyze Dot Product of \(\vec{a}\) and \(\vec{c}\)

Given the additional information that \(\vec{a} \cdot \vec{c} = \frac{1}{2}\), we can conclude that \(\vec{a}\) and \(\vec{c}\) are not orthogonal, hence they have a non-zero projection onto each other, which also implies they cannot be parallel.

Analyzing the statements

Since \(\vec{a} \times \vec{b}\) and \(\vec{c} \times \vec{d}\) both are non-zero and have a dot product equal to 1, we can deduce that: (A) \(\vec{a}, \vec{b}, \vec{c}\) are non-coplanar because \(\vec{a}\) and \(\vec{b}\) form a plane, and \(\vec{a}\) has a non-zero dot product with \(\vec{c}\). (B) \(\vec{b}, \vec{c}, \vec{d}\) are non-coplanar because \(\vec{c}\) and \(\vec{d}\) form a plane, and \(\vec{b}\) is not mentioned to be parallel or orthogonal to \(\vec{c}\) or \(\vec{d}\). (C) \(\vec{b}, \vec{d}\) are non-parallel because their cross product with other vectors results in non-zero vectors. (D) There is no information provided to directly conclude \(\vec{a}, \vec{d}\) are parallel or that \(\vec{b}, \vec{c}\) are parallel, and thus, this statement is incorrect.

Key concepts

Dot Product

Understanding the dot product—or scalar product—is crucial in vector algebra. It's a way to measure how much one vector extends in the direction of another. Mathematically, if we have two vectors, \(\vec{u}\) and \(\vec{v}\), their dot product is defined as \(\vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos(\theta)\), where \(\theta\) is the angle between the vectors and \(|\vec{u}|\) and \(|\vec{v}|\) are their magnitudes.

When the dot product is zero, the vectors are orthogonal, meaning they are at 90 degrees to each other. If the dot product is positive, the angle is less than 90 degrees, and if negative, it's greater than 90 degrees. An intuitive understanding of the dot product is thus essential for analyzing the directional relationship between vectors.

Vector Algebra

Vector algebra encompasses operations involving vectors. It includes addition, subtraction, scaling (multiplication by a scalar), and products (dot and cross). These operations enable us to solve complex problems in physics and engineering related to forces, motion, and fields. For instance, the cross product's result—a vector perpendicular to the original pair—not only states their non-parallelism but also facilitates calculation of torque and angular momentum. The step-by-step solution applied vector algebra by examining the cross and dot products, highlighting its value in determining vector relations in space.

Coplanar Vectors

Vectors are coplanar if they lie in the same plane. This concept is important when working with vector cross products; vectors \(\vec{a}\) and \(\vec{b}\) for example, define a plane and any vector coplanar with them must be linearly dependent. In the exercise, acknowledging that certain groups of vectors were non-coplanar was pivotal. This was inferred through cross products and their associated dot product, which implies vectors \(\vec{a}\) and \(\vec{b}\) are non-parallel and span a plane that \(\vec{c}\) does not lie within, making them non-coplanar.

Parallel and Orthogonal Vectors

Parallel vectors share the same or exactly opposite direction, while orthogonal vectors are perpendicular. These relationships are foundational when analyzing vectorial phenomena. To practice, observe how two parallel vectors have a dot product equal to the product of their magnitudes, and orthogonal vectors have a dot product of zero. The step-by-step solution assigned parallelism and orthogonality based on vector products. For instance, a non-zero cross product definitively means vectors are not parallel. Inversely, a zero cross product indicates parallelism or anti-parallelism. These principles form the heart of vector analysis and are demonstrated by this exercise.