Problem 56 - Precalculus with Calculus Previews

Question

In Problems \(49-62, \mathbf{u}=\langle 2,-3\rangle, \mathbf{v}=\langle-1,5\rangle,\) and \(\mathbf{w}=\langle 3,-2\rangle .\) Find the indicated scalar or vector. $$ \mathbf{v} \cdot(\mathbf{u}-\mathbf{w}) $$

Step-by-Step Solution

Answer

The dot product is \(-4\).

Step 1: Understand the Problem

We need to find the dot product of vector \( \mathbf{v} \) with the difference of vectors \( \mathbf{u} \) and \( \mathbf{w} \). This involves calculating the difference vector first, and then computing the dot product.

Step 2: Compute the Difference Vector

Subtract vector \( \mathbf{w} \) from vector \( \mathbf{u} \). \[ \mathbf{u} - \mathbf{w} = \langle 2, -3 \rangle - \langle 3, -2 \rangle \]Calculate each component: \[ u_1 - w_1 = 2 - 3 = -1 \]\[ u_2 - w_2 = -3 - (-2) = -1 \]So, the difference vector \( \mathbf{u} - \mathbf{w} \) is \( \langle -1, -1 \rangle \).

Step 3: Setup the Dot Product

The dot product \( \mathbf{v} \cdot (\mathbf{u} - \mathbf{w}) \) requires multiplying corresponding components and summing the products. Using vector \( \mathbf{v} = \langle -1, 5 \rangle \) and the difference vector \( \langle -1, -1 \rangle \):\[ \text{Dot Product} = (-1) \times (-1) + 5 \times (-1) \]

Step 4: Calculate the Dot Product

Perform the calculations for the dot product:\[ (-1) \times (-1) = 1 \]\[ 5 \times (-1) = -5 \]Add the results together:\[ 1 + (-5) = -4 \]

Step 5: Conclude the Result

Thus, the result of the dot product \( \mathbf{v} \cdot (\mathbf{u} - \mathbf{w}) \) is \(-4\).

Key Concepts

Understanding Vector Subtraction

Vector subtraction is an operation that allows us to find the difference between two vectors. To subtract vectors, such as our from the example \( \mathbf{u} = \langle 2, -3 \rangle \) and \( \mathbf{w} = \langle 3, -2 \rangle \),we perform component-wise subtraction. For each dimension, we subtract the corresponding components of the second vector from the first.

• For the first component: \( u_1 - w_1 = 2 - 3 = -1 \)
• For the second component: \( u_2 - w_2 = -3 - (-2) = -1 \)

Therefore, the resulting vector from \( \mathbf{u} - \mathbf{w} \) is \( \langle -1, -1 \rangle \).
This process highlights the importance of correctly aligning components during subtraction to avoid errors.By understanding the mechanics of vector subtraction, complex vector operations become more manageable.

Breaking Down Vector Components

Vectors are represented by components that correspond to their dimensions. In a 2-dimensional space, vectors have two components: x and y.This can be seen in vectors such as our given \( \mathbf{v} = \langle -1, 5 \rangle \) where \( -1 \) is the x-component, and \( 5 \) is the y-component.
Components provide a way to calculate and manipulate vectors by performing operations like addition, subtraction, and scaling, on a per-component basis.

• X-component: determines the horizontal influence of the vector.
• Y-component: determines the vertical influence of the vector.

Grasping the concept of vector components simplifies problem-solving in vector mathematics, especially when working with multi-dimensional vectors.

Mastering Vector Operations

Vector operations include the combination and manipulation of vectors to achieve desired outcomes, such as calculating dot products.The dot product is particularly interesting because it results in a scalar rather than a vector. It involves multiplying corresponding components of two vectors and then summing these products.
In our example, the vectors \( \mathbf{v} = \langle -1, 5 \rangle \) and \( \mathbf{u} - \mathbf{w} = \langle -1, -1 \rangle \) are used to perform the dot product operation.

• Multiply corresponding components: \( (-1) \times (-1) = 1 \)
• Multiply next pair of components: \( 5 \times (-1) = -5 \)
• Add the results: \( 1 + (-5) = -4 \)

Understanding such operations allows us to efficiently assess the relationship between vectors in different contexts, like physics or computer graphics.Mastering them greatly improves one's ability to tackle complex vector-based problems.