Question

Let \(\vec{a}, \vec{b}, \vec{c}\) be unit vectors such that \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}\). Which one of the following is correct? (A) \(\vec{a} \times \vec{b}=\vec{b} \times \vec{c}=\vec{c} \times \vec{a}=\overrightarrow{0}\) (B) \(\vec{a} \times \vec{b}=\vec{b} \times \vec{c}=\vec{c} \times \vec{a} \neq \overrightarrow{0}\) (C) \(\vec{a} \times \vec{b}=\vec{b} \times \vec{c}=\vec{a} \times \vec{c} \neq \overrightarrow{0}\) (D) \(\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}\) are mutually perpendicular

Short answer

The correct answer is (D) because the cross products of unit vectors forming an equilateral triangle are mutually perpendicular.

Step-by-step solution

Understand and Analyze the Given Information

Acknowledge that \(\vec{a}, \vec{b}, \vec{c}\) are unit vectors and they sum to the zero vector, \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}\). This implies that each vector is directed towards the negative of the sum of the other two.

Explore Properties of Cross Products

Recall that the cross product of two vectors results in a third vector that is perpendicular to both original vectors, and its magnitude is equal to the area of the parallelogram spanned by the two original vectors. For unit vectors, if the cross product is non-zero, it will be another unit vector perpendicular to the plane formed by the original vectors.

Check Each Option

Examine each given option. For (A), the cross products being zero implies the vectors are parallel, which is not possible given they sum to zero. For (B) and (C), it is not necessary for the cross products of different pairs to be equal. For (D), since the sum of vectors is zero, and they are unit vectors, they form an equilateral triangle in space. The cross products will then indeed be vectors pointing in mutually perpendicular directions.

Key concepts

Understanding Unit Vectors

Unit vectors play a crucial role in vector calculus and physics. A unit vector is a vector that has a magnitude (length) of 1. In other words, it is a vector that points in a specific direction and has a size fixed at one unit. This property makes unit vectors essential for defining direction without the complication of length.

For example, the standard unit vectors in three-dimensional space are denoted by \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \), each pointing in the direction of the x, y, and z-axis respectively. When we talk about \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) being unit vectors, we mean that each of these vectors has an exact length of 1 and specifies a direction in space.

One of the fundamental properties of unit vectors that makes them particularly useful is their ability to serve as building blocks for other vectors through vector addition or scalar multiplication. We can scale them to create vectors of any desired magnitude and in the direction of the original unit vector.

The Principle of Vector Addition

Vector addition is an operation where two or more vectors are combined to form a resultant vector. The process can be visualized by placing the tail of one vector at the head of the preceding vector and then drawing a vector from the tail of the first vector to the head of the last vector. The resultant vector represents both the magnitude and the direction of the total displacement.

In our discussion of unit vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \), they sum up to the zero vector. Mathematically, \( \vec{a} + \vec{b} + \vec{c} = \overrightarrow{0} \). This zero vector is unique because it has no direction and a magnitude of zero, implying that the vectors form a closed path or a triangular loop in space. Since the vectors involved are unit vectors, the triangle is equilateral, suggesting the angular relationship between any two vectors is 120 degrees. This concept is significant when we explore geometric vector operations like the vector cross product.

Calculating Parallelogram Area using Cross Product

The cross product of two vectors \( \vec{A} \times \vec{B} \) results in a new vector that is orthogonal to both \( \vec{A} \) and \( \vec{B} \). Importantly, the magnitude of the cross product vector represents the area of the parallelogram that the original vectors span. This area is calculated as \( |\text{Area}| = |\text{mag}(\vec{A}) \times \text{mag}(\vec{B}) \times \text{sin}(\theta)| \) where \(\theta\) is the angle between \( \vec{A} \) and \( \vec{B} \).

For unit vectors, which have a magnitude of 1, this simplifies the expression, and the magnitude of their cross product tells us the area of the parallelogram is equal to the sine of the angle between them. In the context of the given exercise, where \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) are unit vectors forming an equilateral triangle, the cross products \( \vec{a} \times \vec{b} \) and equivalents represent the areas of parallelograms in space. These areas are essential for visualizing the spatial structures formed by vectors and for solving geometric problems in vector calculus.