Problem 12 - Advanced Engineering Mathematics

Question

In Problems \(1-12, \mathbf{a}=\langle 2,-3,4\rangle, \mathbf{b}=\langle-1,2,5\rangle\), and \(\mathbf{c}=\langle 3,6,-1\rangle .\) Find the indicated scalar or vector. \((\mathbf{c} \cdot \mathbf{b}) \mathbf{a}\)

Step-by-Step Solution

Answer

The result is \( \langle 8, -12, 16 \rangle \).

Step 1: Determine the Dot Product \( \mathbf{c} \cdot \mathbf{b} \)

Find the dot product of vectors \( \mathbf{c} \) and \( \mathbf{b} \) by multiplying their corresponding components and adding the results: \[ (3)(-1) + (6)(2) + (-1)(5) = -3 + 12 - 5 = 4. \]

Step 2: Multiply the Resulting Scalar with Vector \( \mathbf{a} \)

Take the scalar result from the dot product, which is 4, and multiply it by each component of vector \( \mathbf{a} = \langle 2, -3, 4 \rangle \): \[ 4 \cdot \mathbf{a} = 4 \langle 2, -3, 4 \rangle = \langle 4 \times 2, 4 \times (-3), 4 \times 4 \rangle = \langle 8, -12, 16 \rangle. \]

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics. It involves multiplying corresponding components of two vectors and then summing the results. This operation results in a scalar, not a vector. For example, to find the dot product of two vectors \( \mathbf{c} = \langle 3, 6, -1 \rangle \) and \( \mathbf{b} = \langle -1, 2, 5 \rangle \), you multiply each component from \( \mathbf{c} \) with the corresponding component from \( \mathbf{b} \), then add all those products together.

Here's the math for that:

• Multiply the first components: \( 3 \times (-1) = -3 \)
• Multiply the second components: \( 6 \times 2 = 12 \)
• Multiply the third components: \( -1 \times 5 = -5 \)

Finally, add these results to get the dot product: \( -3 + 12 - 5 = 4 \).
This scalar, 4, is the result of the dot product of vectors \( \mathbf{c} \) and \( \mathbf{b} \). It's a simple yet powerful way to multiply vectors!

Scalar Multiplication

Scalar multiplication involves taking a single number (called a scalar) and multiplying it across each component of a vector. This operation stretches or shrinks the vector, depending on the size of the scalar.
For instance, if you have a vector \( \mathbf{a} = \langle 2, -3, 4 \rangle \) and a scalar, let's use 4 from the dot product example, you multiply 4 by each component of vector \( \mathbf{a} \).

Here is how it's done:

• Multiply the scalar with the first component: \( 4 \times 2 = 8 \)
• Multiply the scalar with the second component: \( 4 \times (-3) = -12 \)
• Multiply the scalar with the third component: \( 4 \times 4 = 16 \)

This results in a new vector: \( \langle 8, -12, 16 \rangle \).
The direction of the vector remains the same, but its size is scaled by the scalar. Scalar multiplication is handy for scaling vectors to match certain criteria or for simple linear algebra computations.

Vectors

Vectors are essential mathematical objects used to represent quantities that have both magnitude and direction. They are not just numbers; they tell you both how much and in which direction. Vectors are usually denoted with angled brackets, like \( \langle x, y, z \rangle \) for three-dimensional space. Each component in a vector corresponds to a direction in space.

Key facts about vectors:

• Magnitude: The length or size of the vector, which can be calculated using the Pythagorean theorem for three-dimensional vectors as \( \sqrt{x^2 + y^2 + z^2} \).
• Direction: Expressed relative to a coordinate system, indicating where the vector is pointing.
• Notation: Can be expressed in terms of unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \), which represent the base directions in 3D space.

Vectors are versatile. They are used in physics to represent forces and velocities, in computer graphics for rendering images, and in many other fields where spatial calculations are needed. Understanding vectors is crucial in tackling a wide range of problems, from simple geometry to complex engineering issues. By learning vector operations like the dot product and scalar multiplication, you'll unlock powerful tools for modeling real-world situations.