Problem 16 - Contemporary Precalculus

Question

Find \(u+v, v-u,\) and \(2 u-3 v\). $$\mathbf{u}=\mathbf{i}-\mathbf{j}, \mathbf{v}=2 \mathbf{i}+\mathbf{j}$$

Step-by-Step Solution

Answer

Question: Find the sum (u+v), difference (v-u), and a linear combination (2u-3v) of the given vectors u = i - j and v = 2i + j. Answer: The sum of the vectors is (u+v) = 3i. The difference of the vectors is (v-u) = i + 2j. The linear combination is (2u-3v) = -4i - 5j.

Step 1: Identify the given vectors

We have the vectors u and v: \(\mathbf{u} = \mathbf{i} - \mathbf{j}\) \(\mathbf{v} = 2 \mathbf{i} + \mathbf{j}\)

Step 2: Find the sum (u + v)

To find the sum, we add the corresponding components of the vectors. \((\mathbf{u} + \mathbf{v}) = (\mathbf{i} - \mathbf{j}) + (2 \mathbf{i} + \mathbf{j})\) \((\mathbf{u} + \mathbf{v}) = (1 + 2) \mathbf{i} + (-1 + 1) \mathbf{j}\) \((\mathbf{u} + \mathbf{v}) = 3\mathbf{i} + 0\mathbf{j}\)

Step 3: Find the difference (v - u)

To find the difference, we subtract the corresponding components of the vectors. \((\mathbf{v} - \mathbf{u}) = (2 \mathbf{i} + \mathbf{j}) - (\mathbf{i} - \mathbf{j})\) \((\mathbf{v} - \mathbf{u}) = (2 - 1) \mathbf{i} + (1 + 1) \mathbf{j}\) \((\mathbf{v} - \mathbf{u}) = \mathbf{i} + 2\mathbf{j}\)

Step 4: Find the linear combination (2u - 3v)

To find the linear combination, we multiply the vectors by the given scalars and then subtract the corresponding components of the resulting vectors. \((2\mathbf{u} - 3\mathbf{v}) = 2(\mathbf{i} - \mathbf{j}) - 3(2 \mathbf{i} + \mathbf{j})\) \((2\mathbf{u} - 3\mathbf{v}) = (2 \mathbf{i} - 2 \mathbf{j}) - (6 \mathbf{i} + 3 \mathbf{j})\) \((2\mathbf{u} - 3\mathbf{v}) = (2 - 6) \mathbf{i} + (-2 - 3) \mathbf{j}\) \((2\mathbf{u} - 3\mathbf{v}) = -4 \mathbf{i} - 5 \mathbf{j}\) So, the final results are: \((\mathbf{u} + \mathbf{v}) = 3\mathbf{i}\) \((\mathbf{v} - \mathbf{u}) = \mathbf{i} + 2\mathbf{j}\) \((2\mathbf{u} - 3\mathbf{v}) = -4 \mathbf{i} - 5 \mathbf{j}\)

Key Concepts

Vector Subtraction

Vector subtraction helps us find the difference between two vectors. Imagine vectors as arrows in space. To subtract one vector from another, we can think of it as finding the change needed to go from one vector to the other. In our example, for the vectors \( \mathbf{u} = \mathbf{i} - \mathbf{j} \) and \( \mathbf{v} = 2\mathbf{i} + \mathbf{j} \), subtracting \( \mathbf{u} \) from \( \mathbf{v} \) involves:

• Identifying the components: \( \mathbf{v} \) has components \( 2 \mathbf{i} \) and \( \mathbf{j} \), while \( \mathbf{u} \) has components \( \mathbf{i} \) and \(-\mathbf{j} \).
• Subtract the corresponding components: \( (2 - 1) \mathbf{i} + (1 + 1) \mathbf{j} \).
• Result: \( \mathbf{i} + 2 \mathbf{j} \).

The subtraction results in a vector pointing from the tip of \( \mathbf{u} \) to the tip of \( \mathbf{v} \), giving us a clear visualization of the difference between the two vectors.

Linear Combinations

Linear combinations allow us to create new vectors by adding together multiples of other vectors. It is like mixing different ingredients to create a new recipe. Let's look at the combination \( 2 \mathbf{u} - 3 \mathbf{v} \):

• Multiply: Scale \( \mathbf{u} = \mathbf{i} - \mathbf{j} \) by 2 to get \( 2\mathbf{i} - 2\mathbf{j} \).
• Scale \( \mathbf{v} = 2\mathbf{i} + \mathbf{j} \) by 3 to have \( 6\mathbf{i} + 3\mathbf{j} \).
• Subtract: \( (2\mathbf{i} - 2\mathbf{j}) - (6\mathbf{i} + 3\mathbf{j}) \).
• Simplify: \((2 - 6)\mathbf{i} + (-2 - 3)\mathbf{j} = -4 \mathbf{i} - 5 \mathbf{j} \).

The linear combination creates a vector involving contributions from both vectors \( \mathbf{u} \) and \( \mathbf{v} \), but scaled by the amounts we choose. It shows how versatile vectors can be, shaping them into whatever form we need for a solution.

Coordinate Vectors

Coordinate vectors help us understand vectors via their individual components in a coordinate plane. Each vector can be broken down into parts along the \( \mathbf{i} \) and \( \mathbf{j} \) directions, forming a "coordinate vector" that simplifies operations:

• \( \mathbf{u} \) can be represented as \( (1, -1) \) in i and j directions.
• \( \mathbf{v} \) appears as \( (2, 1) \).
• Add them: resulting in the coordinate vector \( (3, 0) \) for \( \mathbf{u} + \mathbf{v} \).
• Subtract: leads to \( (1, 2) \) for \( \mathbf{v} - \mathbf{u} \).

By breaking vectors down to these components, it's easier to perform arithmetic operations. Coordinate vectors offer a structured way to see how individual parts of a vector interact in calculations and help in understanding geometric concepts.