Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). \(\operatorname{proj}_{\mathbf{u}}(\mathbf{w}-\mathbf{v})\)

Step-by-Step Solution

Answer

The projection is \( \frac{11}{5} \mathbf{i} + \frac{22}{5} \mathbf{j} \).

Step 1: Find \( \mathbf{w} - \mathbf{v} \)

To find \( \mathbf{w} - \mathbf{v} \), subtract the components of \( \mathbf{v} \) from \( \mathbf{w} \): \[ \mathbf{w} - \mathbf{v} = (1 \mathbf{i} + 5 \mathbf{j}) - (2 \mathbf{i} - 1 \mathbf{j}) \]This results in:\[\mathbf{w} - \mathbf{v} = (1 - 2) \mathbf{i} + (5 + 1) \mathbf{j} = -1 \mathbf{i} + 6 \mathbf{j}\].

Step 2: Calculate Dot Product \( \mathbf{u} \cdot (\mathbf{w} - \mathbf{v}) \)

Now compute the dot product of \( \mathbf{u} \) and \( \mathbf{w} - \mathbf{v} \):\[\mathbf{u} = \mathbf{i} + 2\mathbf{j}\]\[\mathbf{w} - \mathbf{v} = -\mathbf{i} + 6\mathbf{j}\]\( \mathbf{u} \cdot (\mathbf{w} - \mathbf{v}) = (1)(-1) + (2)(6) = -1 + 12 = 11 \).

Step 3: Calculate Dot Product \( \mathbf{u} \cdot \mathbf{u} \)

Compute the dot product of \( \mathbf{u} \) with itself:\[\mathbf{u} \cdot \mathbf{u} = (1 \cdot 1) + (2 \cdot 2) = 1 + 4 = 5.\]

Step 4: Find the Projection \( \operatorname{proj}_{\mathbf{u}}(\mathbf{w}-\mathbf{v}) \)

The projection formula is given by:\[ \operatorname{proj}_{\mathbf{u}}(\mathbf{w} - \mathbf{v}) = \frac{\mathbf{u} \cdot (\mathbf{w} - \mathbf{v})}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u} \].Substitute the computed values:\[ \operatorname{proj}_{\mathbf{u}}(\mathbf{w} - \mathbf{v}) = \frac{11}{5}(\mathbf{i} + 2\mathbf{j}). \]Thus:\[ \operatorname{proj}_{\mathbf{u}}(\mathbf{w} - \mathbf{v}) = \left(\frac{11}{5}\right)\mathbf{i} + \left(\frac{22}{5}\right)\mathbf{j}. \]

Key Concepts

Dot Product

The dot product is a key operation when working with vectors. It combines two vectors to produce a scalar. This scalar helps determine the relationship between the vectors, such as how much one vector influences the direction of another. To compute the dot product of two vectors, like \( \mathbf{u} \) and \( \mathbf{a} \), you'll follow these steps:

• Multiply the corresponding components of the vectors together.
• Add the results of these multiplications.

For example, given vectors \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) and\( \mathbf{a} = c\mathbf{i} + d\mathbf{j} \):\[\mathbf{u} \cdot \mathbf{a} = (a \cdot c) + (b \cdot d)\]In the original exercise, the dot product \( \mathbf{u} \cdot (\mathbf{w} - \mathbf{v}) \) is important for understanding how the vectors align or differ.

Vector Subtraction

Vector subtraction is used to find the difference between two vectors. It is crucial when you are interested in how one vector changes in relation to another. The operation is straightforward: subtract each component of one vector from the corresponding component of the other vector. Suppose you have vectors \( \mathbf{w} \) and \( \mathbf{v} \). To find \( \mathbf{w} - \mathbf{v} \), perform the following:

• Subtract the \( i \)-components of \( \mathbf{v} \) from \( \mathbf{w} \)
• Subtract the \( j \)-components of \( \mathbf{v} \) from \( \mathbf{w} \)

Mathematically,\[\mathbf{w} - \mathbf{v} = (w_1 - v_1)\mathbf{i} + (w_2 - v_2)\mathbf{j}\]In the provided problem, this step yields vector \(-\mathbf{i} + 6\mathbf{j}\). This vector represents the directional difference between \( \mathbf{w} \) and \( \mathbf{v} \).

Projection Formula

The projection formula is essential for determining how much one vector points in the direction of another. It calculates a vector projection, producing a vector that shows how much of one vector lies along another. The projection of vector \( \mathbf{a} \) onto vector \( \mathbf{b} \) is given by:\[\operatorname{proj}_{\mathbf{b}}(\mathbf{a}) = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \right) \mathbf{b}\]This formula uses the dot product to measure alignment and scales vector\( \mathbf{b} \) accordingly. In the exercise, we applied it to project \( \mathbf{w} - \mathbf{v} \) onto \( \mathbf{u} \). This shows how much of the resultant vector from subtraction points in the direction of \( \mathbf{u} \).In practice, the formula determines how aligned, or even coplanar, two vectors are.

Vector Components

Understanding vector components is fundamental in vector mathematics. Vectors are often broken down into their components along the \( i \) and \( j \) directions, making it easier to perform calculations like dot products and subtractions.Each vector can be expressed in terms of its components: \( \mathbf{v} = x\mathbf{i} + y\mathbf{j} \). Here, \( x \) and \( y \) are the vector's components along the Cartesian coordinate axes.Key to:

• Adding and subtracting vectors component-wise.
• Calculating dot products, which involve multiplying and summing corresponding components.

Recognizing components allows you to break down more complex vectors into manageable parts. When working through steps in vector projections, being clear about each component's role allows for precision and clarity in problem-solving.