Problem 39 - Algebra and Trigonometry

Question

Let $$\mathbf{u}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}, \quad \text { and } \quad \mathbf{w}=-5 \mathbf{j}$$ Find each specified scalar or vector. $$ 5 \mathbf{u} \cdot(3 \mathbf{v}-4 \mathbf{w}) $$

Step-by-Step Solution

Answer

The result of the dot product \( 5u \cdot (3v-4w) \) is 25.

Step 1: Calculate 3v - 4w

Multiply the scalar values with the corresponding vectors to get a new resulting vector. That is, 3v - 4w = \( (3*3\mathbf{i}-2*3\mathbf{j}) - (4*(-5\mathbf{j})) = (9\mathbf{i}-6\mathbf{j} + 20\mathbf{j}) = 9\mathbf{i} + 14\mathbf{j} \).

Step 2: Multiply 5u with the result from Step 1

The given exercise asks for the dot product of 5u and (3v - 4w). We have to multiply 5u with the resulting vector from Step 1. Calculation: \( 5u = 5(-\mathbf{i} + \mathbf{j}) = -5\mathbf{i} + 5\mathbf{j} \).

Step 3: Compute the dot product

The dot product of two vectors \( \mathbf{a} = a_{1}\mathbf{i} + a_{2}\mathbf{j} \) and \( \mathbf{b} = b_{1}\mathbf{i} + b_{2}\mathbf{j} \) is calculated as \( \mathbf{a} \cdot \mathbf{b} = a_{1}b_{1} + a_{2}b_{2} \). Applying this, the dot product of the vectors from Step 2: \( (-5\mathbf{i} + 5\mathbf{j}) \cdot (9\mathbf{i} + 14\mathbf{j}) = -5*9 + 5*14 = -45 + 70 = 25 \).

Key Concepts

Understanding the Dot Product

The dot product is an operation that takes two vectors and returns a single number, or scalar. This operation is critical in various fields like physics and computer graphics because it gives an idea of how much one vector is aligned with another. It works by multiplying corresponding components of the vectors and then summing the products.

To visualize, consider vectors \( \mathbf{a} = a_{1}\mathbf{i} + a_{2}\mathbf{j} \) and \( \mathbf{b} = b_{1}\mathbf{i} + b_{2}\mathbf{j} \). The dot product can be expressed as:

• \( \mathbf{a} \cdot \mathbf{b} = a_{1}b_{1} + a_{2}b_{2} \)

This is a simple arithmetic operation. Align the components and multiply them correspondingly.

• If the vectors are perpendicular, the dot product is zero.
• If the vectors face in the same direction, the product is positive.
• If they are in opposite directions, it is negative.

Scalar Multiplication and Its Application

Scalar multiplication involves multiplying a vector by a scalar, or single number, which effectively scales the vector by stretching or compressing it. For example, multiplying by a scalar greater than one lengthens the vector, while a scalar less than one shortens it.

Suppose you have vector \( \mathbf{u} = -\mathbf{i} + \mathbf{j} \). If you multiply it by 5, you scale both its components:

• \( 5\mathbf{u} = 5(-\mathbf{i} + \mathbf{j}) = -5\mathbf{i} + 5\mathbf{j} \)

After multiplication, the direction remains unchanged, but the vector's magnitude alters proportionally. This operation is straightforward but integral to many vector processes like finding the dot product or analyzing physical systems within linear equations.

Exploring Vector Addition

Vector addition is the process of combining two or more vectors to form a new resultant vector. This is done by adding corresponding components of the vectors. For example, adding \( \mathbf{a} \) and \( \mathbf{b} \) is simply combining each directional piece.

• If \( \mathbf{a} = a_{1}\mathbf{i} + a_{2}\mathbf{j} \) and \( \mathbf{b} = b_{1}\mathbf{i} + b_{2}\mathbf{j} \), then \( \mathbf{a} + \mathbf{b} = (a_{1} + b_{1})\mathbf{i} + (a_{2} + b_{2})\mathbf{j} \).

As illustrated in the exercise, we added vectors after scaling them with the scalars 3 and 4.

• First, compute \( 3\mathbf{v} - 4\mathbf{w} = 9\mathbf{i} - 6\mathbf{j} + 20\mathbf{j} = 9\mathbf{i} + 14\mathbf{j} \).

This reflects the straightforward nature of vector addition, a crucial skill in linear algebra and beyond.

A Brief Journey into Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It is the language through which much of modern physics, engineering, and economics is expressed.

At its core, linear algebra concerns itself with vectors and how they interact through operations like dot products, scalar multiplications, and additions. It provides a framework to model systems and solve them efficiently. Linear algebra forms the basis for fields like quantum mechanics and statistics.

• For instance, in the exercise we explored, you encountered basic vector operations, realizing the essence of linear algebra.
• It lets us articulate complex 3D spaces and transformations elegantly and succinctly.

Therefore, understanding these concepts gives you tools crucial for delving into advanced mathematical theories and solving real-world problems.