Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find each of the given projections if \(\mathbf{u}=3 \mathbf{i}+2 \mathbf{j}+\mathbf{k}, \mathbf{v}=2 \mathbf{i}-\mathbf{k}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}-3 \mathbf{k}\). \(\operatorname{proj}_{\mathbf{u}}(\mathbf{w}+\mathbf{v})\)

Step-by-Step Solution

Answer

The projection is \( \frac{45}{14}\mathbf{i} + \frac{30}{14}\mathbf{j} + \frac{15}{14}\mathbf{k} \).

Step 1: Find the vector sum \( \mathbf{w} + \mathbf{v} \)

First, we need to find the vector \( \mathbf{w} + \mathbf{v} \). Given \( \mathbf{w} = \mathbf{i} + 5 \mathbf{j} - 3 \mathbf{k} \) and \( \mathbf{v} = 2 \mathbf{i} - \mathbf{k} \), add them component-wise.\[ (\mathbf{i} + 5\mathbf{j} - 3\mathbf{k}) + (2\mathbf{i} - \mathbf{k}) = (1+2)\mathbf{i} + 5 \mathbf{j} + (-3-1)\mathbf{k} = 3\mathbf{i} + 5\mathbf{j} - 4\mathbf{k} \]. Consequently, \( \mathbf{w} + \mathbf{v} = 3\mathbf{i} + 5\mathbf{j} - 4\mathbf{k}. \)

Step 2: Calculate the dot product \( \mathbf{u} \cdot (\mathbf{w} + \mathbf{v}) \)

Now, calculate the dot product of \( \mathbf{u} = 3\mathbf{i} + 2\mathbf{j} + \mathbf{k} \) and \( \mathbf{w} + \mathbf{v} = 3\mathbf{i} + 5\mathbf{j} - 4\mathbf{k} \). Use the formula: \( \mathbf{u} \cdot \mathbf{b} = u_1b_1 + u_2b_2 + u_3b_3 \).\[ (3)(3) + (2)(5) + (1)(-4) = 9 + 10 - 4 = 15. \] Therefore, \( \mathbf{u} \cdot (\mathbf{w} + \mathbf{v}) = 15 \).

Step 3: Calculate \( |\mathbf{u}|^2 \)

Calculate the magnitude squared of \( \mathbf{u} = 3\mathbf{i} + 2\mathbf{j} + \mathbf{k} \). To find \( |\mathbf{u}|^2 \), use the formula \( |\mathbf{u}|^2 = u_1^2 + u_2^2 + u_3^2 \).\[ 3^2 + 2^2 + 1^2 = 9 + 4 + 1 = 14. \] Hence, \( |\mathbf{u}|^2 = 14 \).

Step 4: Compute the projection \( \operatorname{proj}_{\mathbf{u}}(\mathbf{w}+\mathbf{v}) \)

Use the formula for the projection of vector \( \mathbf{b} \) onto vector \( \mathbf{a} \): \( \operatorname{proj}_{\mathbf{a}} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|^2} \mathbf{a} \). Substitute \( \mathbf{u} \cdot (\mathbf{w} + \mathbf{v}) = 15 \) and \( |\mathbf{u}|^2 = 14 \) into this formula.\[ \operatorname{proj}_{\mathbf{u}}(\mathbf{w}+\mathbf{v}) = \frac{15}{14}(3\mathbf{i} + 2\mathbf{j} + \mathbf{k}) \]. Therefore, the projection is \( \frac{15}{14} \times 3\mathbf{i} + \frac{15}{14} \times 2\mathbf{j} + \frac{15}{14} \times \mathbf{k} = \frac{45}{14}\mathbf{i} + \frac{30}{14}\mathbf{j} + \frac{15}{14}\mathbf{k} \).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It combines two vectors and outputs a scalar. The dot product between two vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \) is calculated using the formula: \[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \] This operation multiplies each corresponding component and then sums these products. It gives insight into the relationship between the vectors, with the resulting scalar indicating both the magnitude of the vectors and how aligned they are.- A positive dot product implies the vectors are pointing in a generally similar direction.- A zero dot product indicates they are perpendicular.- A negative result signifies they point in opposite directions.The dot product reflects the degree to which two vectors are aligned. In the solution, the dot product of \( \mathbf{u} \) and \( \mathbf{w} + \mathbf{v} \) revealed a value of 15, showing they share a certain level of directionality.

Vector Addition

Vector addition is a straightforward method to combine two vectors and results in a new vector. This process involves aligning the tail of one vector to the head of another and then determining the resulting vector from this configuration.When you add two vectors, as in the case of \( \mathbf{w} \) and \( \mathbf{v} \), it is done by simply summing their corresponding components: \[ (\mathbf{i} + 5\mathbf{j} - 3\mathbf{k}) + (2\mathbf{i} - \mathbf{k}) = (1+2)\mathbf{i} + 5 \mathbf{j} + (-3-1)\mathbf{k} = 3\mathbf{i} + 5\mathbf{j} - 4\mathbf{k} \]The resultant vector \( \mathbf{w} + \mathbf{v} \) expresses how you would "travel" through space if you took the path indicated by \( \mathbf{w} \) followed directly by \( \mathbf{v} \). This technique not only simplifies complex vector routes but is also foundational for further vector operations, like projections.

Magnitude of a Vector

The magnitude of a vector can be thought of as its "length" in space. It shows how much the vector "stretches" from its starting point to its endpoint. For a vector \( \mathbf{u} = u_1\mathbf{i} + u_2\mathbf{j} + u_3\mathbf{k} \), its magnitude is calculated using the formula: \[ |\mathbf{u}| = \sqrt{u_1^2 + u_2^2 + u_3^2} \]This calculation involves taking the square root of the sum of the squares of its components. However, when dealing with vector projections, you often need the magnitude squared, \( |\mathbf{u}|^2 \), which is simply without the square root: \[ |\mathbf{u}|^2 = u_1^2 + u_2^2 + u_3^2 \]In our context, we found \( |\mathbf{u}|^2 = 14 \) for the vector \( \mathbf{u} \). This information is crucial because it normalizes the direction of the vector during projection.

Component-wise Addition

Component-wise addition is the method of adding vectors by summing each corresponding component individually. It's akin to solving almost three small additions independently and is essential in vector operations such as calculating the sum of vectors or working on projections.Consider vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \). To find \( \mathbf{a} + \mathbf{b} \), simply add each pair of corresponding components:\[ \mathbf{a} + \mathbf{b} = (a_1+b_1)\mathbf{i} + (a_2+b_2)\mathbf{j} + (a_3+b_3)\mathbf{k} \]In our case, using this method, we calculated \( \mathbf{w} + \mathbf{v} = 3\mathbf{i} + 5\mathbf{j} - 4\mathbf{k} \). This component-wise strategy is fundamental and very helpful for visualizing and performing vector operations in physics and engineering settings.