Problem 2 - Precalculus Mathematics for Calculus

Question

The cross product of two vectors a and \(\mathbf{b}\) is ___________________ to \(\mathbf{a}\) and to \(\mathbf{b}\). Thus if both vectors a and b lie in a plane, the vector \(\mathbf{a} \times \mathbf{b}\) is _________________ to the plane.

Step-by-Step Solution

Answer

perpendicular; perpendicular

Step 1: Understanding the Cross Product

The cross product of two vectors, \( \mathbf{a} \times \mathbf{b} \), results in a vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). This property can be leveraged to determine the orientation of the resultant vector compared to the original vectors.

Step 2: Analyzing the Plane

When vectors \( \mathbf{a} \) and \( \mathbf{b} \) lie within the same plane, the plane itself is defined by all linear combinations of these two vectors. Hence, the cross product provides information about direction relative to this plane.

Step 3: Identifying the Resultant Direction

Since the cross product \( \mathbf{a} \times \mathbf{b} \) is orthogonal to both \( \mathbf{a} \) and \( \mathbf{b} \), it is by definition also orthogonal to any plane formed by these vectors. Thus, the resultant vector \( \mathbf{a} \times \mathbf{b} \) is perpendicular to the plane that contains \( \mathbf{a} \) and \( \mathbf{b} \).

Key Concepts

Perpendicular Vectors

When we talk about perpendicular vectors in the context of the vector cross product, we are referring to a special relationship between two vectors and their resultant vector. The cross product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \), symbolized as \( \mathbf{a} \times \mathbf{b} \), leads to a third vector that has a unique property: it is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \).

• The perpendicular nature means that the angle between the resultant vector and both initial vectors is 90 degrees, often described using the term 'orthogonal'.
• This orthogonal vector indicates the direction that is not aligned with either of the two vectors being multiplied, creating a useful tool for defining spatial relationships in three dimensions.

Understanding this concept is crucial, as it lays the groundwork for geometric and physical applications where directionality and alignment are often more essential than mere magnitude.

Plane Geometry

In plane geometry, vectors play a pivotal role in describing directions and boundaries within a plane. When two vectors, \( \mathbf{a} \) and \( \mathbf{b} \), lie within the same plane, they essentially define the plane's orientation and span.

• Any vector that resides in the same plane can be expressed as a linear combination of \( \mathbf{a} \) and \( \mathbf{b} \).
• This means that by adjusting the coefficients of \( \mathbf{a} \) and \( \mathbf{b} \), you can reach any point within that planar space.
• The defined plane serves to help visualize how these vectors interact and where the resulting vector from their cross product will sit relative to the original plane.

Thus, the concept of the plane is not only a mere mathematical abstraction but a tool for understanding and solving geometric problems involving interactions of forces, directions, and spaces.

Resultant Vector Direction

The resultant vector direction is a defining characteristic of the vector cross product. When you perform a cross product on two vectors, such as \( \mathbf{a} \times \mathbf{b} \), the resulting vector does not lie in the same plane as \( \mathbf{a} \) and \( \mathbf{b} \). Here's how it works:

• The resultant vector is orthogonal to the plane established by \( \mathbf{a} \) and \( \mathbf{b} \), which means it points "out of the plane."
• To determine the direction of this vector, the right-hand rule is often used. To apply it, point your fingers in the direction of \( \mathbf{a} \) and curl them towards \( \mathbf{b} \); your thumb will then indicate the direction of \( \mathbf{a} \times \mathbf{b} \).
• This directional property is exceedingly practical in physics and engineering for indicating torque, rotational forces, or any vector quantities that require a distinct perpendicular orientation relative to a defined set of initial vectors.

The resultant vector's direction clarifies how vector orientations relate to each other and how they might affect motion or mechanical systems in space.