Problem 14 - Advanced Engineering Mathematics

Question

Find a vector that is perpendicular to both a and b. $$ \mathbf{a}=(-1,-2,4), \mathbf{b}=\langle 4,-1,0\rangle $$

Step-by-Step Solution

Answer

The vector perpendicular to both is \((4, 16, 9)\).

Step 1: Understand the Concept of Cross Product

The cross product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) results in a vector \(\mathbf{c}\) that is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\). We will calculate the cross product \(\mathbf{a} \times \mathbf{b}\) to find such a vector.

Step 2: Use the Formula for Cross Product

The formula for the cross product \(\mathbf{a} \times \mathbf{b}\) is given by: \[\mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)\] We substitute the values from \(\mathbf{a}=(-1,-2,4)\) and \(\mathbf{b}=(4,-1,0)\).

Step 3: Calculate the Individual Components

Calculate each component of the cross product using the formula:\(a_2b_3 - a_3b_2 = (-2)(0) - (4)(-1) = 0 + 4 = 4a_3b_1 - a_1b_3 = (4)(4) - (-1)(0) = 16 + 0 = 16a_1b_2 - a_2b_1 = (-1)(-1) - (-2)(4) = 1 + 8 = 9\)

Key Concepts

Perpendicular Vectors and Their Cross Product

Perpendicular vectors are vectors that intersect at a right angle, forming a 90-degree angle between them. This concept is crucial in vector algebra, particularly when working in three-dimensional space. An important operation used to find a vector perpendicular to two given vectors is the **cross product**. If you have two vectors, \( \mathbf{a} \) and \( \mathbf{b} \), the cross product \( \mathbf{a} \times \mathbf{b} \) results in a third vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). This resulting vector is often denoted as \( \mathbf{c} \). In practical terms, if you imagine \( \mathbf{a} \) and \( \mathbf{b} \) laying on a plane, the cross product vector will "stick out" of the plane.

• If a vector \( \mathbf{c} \) is perpendicular to any vectors \( \mathbf{a} \) and \( \mathbf{b} \), it satisfies: \( \mathbf{a} \cdot \mathbf{c} = 0 \) and \( \mathbf{b} \cdot \mathbf{c} = 0 \)

Understanding this helps in many applications, from computer graphics to physics, where notions of right angles and perpendicularity are integral.

Basics of Vector Algebra: Understanding Vector Operations

Vector algebra deals with operations involving vectors, critical in fields like physics and engineering and when working with three-dimensional vectors. A vector is essentially a quantity that has both a magnitude and a direction, represented as an ordered list of numbers, like \((x, y, z)\). Two principal operations in vector algebra are the **dot product** and the **cross product**.

• The dot product is a scalar result obtained from two vectors. It tells you how much of one vector goes in the direction of another. A result of zero implies the vectors are perpendicular.
• The cross product results in a vector, important for determining perpendicular vectors as we've seen earlier.

Moreover, knowing how to properly execute additions, subtractions, and the scalar multiplication of vectors plays a crucial role in manipulating and understanding vector quantities and their properties in various dimensions.

Three-Dimensional Vectors: Navigating 3D Space

Three-dimensional vectors are vectors that exist in space with three distinct dimensions—height, width, and depth, often denoted as \( x, y, z \). In the context of 3D space, vectors are especially useful for modeling and solving problems related to physical phenomena, like force, velocity, and acceleration.Some useful pointers for understanding 3D vectors include:

• A 3D vector can be represented as \( \langle x, y, z \rangle \), which indicates its position or direction in 3D space.
• Operations on 3D vectors, such as addition, subtraction, and cross product, maintain their three-dimensional nature.
• Visualizing 3D vectors often involves using a coordinate system where each axis represents one dimension.

These vectors become extraordinarily potent tools in vector calculus and physics, as they help describe and predict the behavior of physical systems using axes like the Cartesian coordinate system.