Problem 24 - Contemporary Precalculus

Question

Find the components of the given vector, where \(\boldsymbol{u}=\boldsymbol{i}-2 \boldsymbol{j}, \boldsymbol{v}=3 \boldsymbol{i}+\boldsymbol{j}, \boldsymbol{w}=-4 \boldsymbol{i}+\boldsymbol{j}\) $$-2 \mathbf{u}+3 \mathbf{v}$$

Step-by-Step Solution

Answer

Question: Find the components of the vector resulting from the linear combination of the given vectors $$\mathbf{u}=\mathbf{i}-2\mathbf{j}$$ and $$\mathbf{v}=3\mathbf{i}+\mathbf{j}$$ using the expression $$-2\mathbf{u}+3\mathbf{v}$$. Answer: The components of the vector resulting from the linear combination of the given vectors are: $$-2\mathbf{u}+3\mathbf{v} = 7\mathbf{i}+7\mathbf{j}$$.

Step 1: Multiply the vectors by their coefficients.

First, we need to multiply the vectors \(\mathbf{u}\) and \(\mathbf{v}\) by their respective coefficients (-2 and 3): $$-2\mathbf{u} = -2(\mathbf{i} - 2\mathbf{j}) = -2\mathbf{i} + 4\mathbf{j}$$ $$3\mathbf{v} = 3(3\mathbf{i} + \mathbf{j}) = 9\mathbf{i} + 3\mathbf{j}$$

Step 2: Calculate the sum of the vectors.

Next, we need to find the sum of the vectors obtained in the previous step: $$-2\mathbf{u} + 3\mathbf{v} = (-2\mathbf{i} + 4\mathbf{j}) + (9\mathbf{i} + 3\mathbf{j})$$

Step 3: Add the components.

Lastly, we need to add the components of the vectors, i.e., add the \(\mathbf{i}\) components and the \(\mathbf{j}\) components separately: $$(-2\mathbf{i} + 4\mathbf{j}) + (9\mathbf{i} + 3\mathbf{j}) = (-2 + 9)\mathbf{i} + (4 + 3)\mathbf{j} = 7\mathbf{i} + 7\mathbf{j}$$ Now, we have the components of the vector resulting from the linear combination of \(\mathbf{u}\) and \(\mathbf{v}\): $$(-2 \mathbf{u}+3 \mathbf{v})_\mathbf{i} = 7$$ $$(-2 \mathbf{u}+3 \mathbf{v})_\mathbf{j} = 7$$ So the components of the given vector are: $$-2 \mathbf{u}+3 \mathbf{v} = 7\mathbf{i} + 7\mathbf{j}$$

Key Concepts

Vector Components

In the world of mathematics, vectors are quantities that have both magnitude and direction, represented in terms of their components. When we talk about vector components, we refer to the representation of a vector in terms of its basis vectors, often denoted as \(\mathbf{i}\) and \(\mathbf{j}\) in two-dimensional space. These basis vectors typically represent the horizontal and vertical directions respectively.

For instance, the vector \(\mathbf{u} = \mathbf{i} - 2\mathbf{j}\) has components 1 in the \(\mathbf{i}\) direction and -2 in the \(\mathbf{j}\) direction. Similarly, \(\mathbf{v} = 3\mathbf{i} + \mathbf{j}\) has components 3 and 1. Understanding these components is essential because they allow us to perform mathematical operations such as addition, subtraction, or scaling easily by working with simple numbers rather than complex directional entities.

Components give us a structured way to express vectors such that operations involving different vectors become straightforward. If you think of a vector as an arrow pointing in a particular direction, its components tell you exactly how far the arrow reaches horizontally and vertically from its starting point. This makes understanding vector interactions much simpler, as demonstrated when finding vector sums or products.

Linear Combination

One of the foundational tools in working with vectors is the concept of a linear combination. This involves creating new vectors from existing ones by multiplying each vector by a scalar (a constant number) and then adding the results together.

In the given example, we computed \(-2 \mathbf{u} + 3 \mathbf{v}\). Here, \(\mathbf{u}\) and \(\mathbf{v}\) are the original vectors, and \(-2\) and \(3\) are the scalars or coefficients. The linear combination process begins with scaling each vector by its respective scalar:
\(-2 \mathbf{u} = -2(\mathbf{i} - 2\mathbf{j})\) and \(3 \mathbf{v} = 3(3\mathbf{i} + \mathbf{j})\). Once scaled, we simply add the results: \((-2\mathbf{i} + 4\mathbf{j}) + (9\mathbf{i} + 3\mathbf{j})\).

Linear combinations are powerful because they allow us to express complex movements or transformations as combinations of simpler, known vectors. This concept is extensively used in fields such as physics, engineering, and computer graphics where vector operations are frequent. By grasping the idea of a linear combination, one gains a crucial tool for manipulating and understanding vector spaces.

Vector Addition

Vector addition involves adding the corresponding components of two or more vectors. It's a process with deep implications and intuitive outcomes. When adding vectors, you're essentially determining where the endpoint of the resultant vector will land given the displacements caused by each vector involved.

Take the vectors \(-2 \mathbf{u} = -2\mathbf{i} + 4\mathbf{j}\) and \(3 \mathbf{v} = 9\mathbf{i} + 3\mathbf{j}\) from the exercise. Adding these vectors means:

• Add the \(\mathbf{i}\) components: \(-2 + 9 = 7\)
• Add the \(\mathbf{j}\) components: \(4 + 3 = 7\)

Thus, \(-2 \mathbf{u} + 3 \mathbf{v} = 7\mathbf{i} + 7\mathbf{j}\).

The beauty of vector addition lies in its versatility. It's akin to finding out the result of combining different routes or paths. Visually, you could represent it with arrows: place the tail of the second arrow at the head of the first, and the resultant vector runs directly from the start of the first vector to the endpoint of the second. Accurate vector addition is crucial in applications like navigation systems, robotics, and physics simulations, where precise calculations of paths and directions are necessary.