Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the scalar projection of \(\mathbf{u}=5 \mathbf{i}+5 \mathbf{j}+2 \mathbf{k}\) on \(\mathbf{v}=-\sqrt{5} \mathbf{i}+\sqrt{5} \mathbf{j}+\mathbf{k}\)

Step-by-Step Solution

Answer

The scalar projection of \(\mathbf{u}\) on \(\mathbf{v}\) is \(\frac{2\sqrt{11}}{11}\).

Step 1: Write the Formula for Scalar Projection

The scalar projection of a vector \(\mathbf{u}\) on \(\mathbf{v}\) is given by the formula:\[ \text{Scalar Projection of } \mathbf{u} ext{ on } \mathbf{v} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|} \]where \(\mathbf{u} \cdot \mathbf{v}\) is the dot product of \(\mathbf{u}\) and \(\mathbf{v}\), and \(\|\mathbf{v}\|\) is the magnitude of \(\mathbf{v}\).

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

The dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is given by:\[ \mathbf{u} \cdot \mathbf{v} = (5)(-\sqrt{5}) + (5)(\sqrt{5}) + (2)(1) \]Calculating this, we have:\[ 5(-\sqrt{5}) + 5(\sqrt{5}) + 2 = -5\sqrt{5} + 5\sqrt{5} + 2 = 2\]

Step 3: Calculate the Magnitude \( \|\mathbf{v}\| \)

The magnitude of \(\mathbf{v}\) is calculated as:\[ \|\mathbf{v}\| = \sqrt{((-\sqrt{5})^2) + ((\sqrt{5})^2) + (1)^2} \]\[ = \sqrt{5 + 5 + 1} = \sqrt{11} \]

Step 4: Compute the Scalar Projection

Now, using the values from the previous steps, compute the scalar projection:\[ \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|} = \frac{2}{\sqrt{11}} \]

Step 5: Simplify the Scalar Projection

The scalar projection can also be expressed in a simplified form by rationalizing the denominator:\[ \frac{2}{\sqrt{11}} \times \frac{\sqrt{11}}{\sqrt{11}} = \frac{2\sqrt{11}}{11} \]

Key Concepts

Dot Product

The dot product is a way to multiply two vectors, resulting in a scalar. It's an essential operation in vector algebra. To calculate the dot product of two vectors, you multiply corresponding components and then add those products together. The formula looks like this:

• For vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \), the dot product is \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).

In the example given, the dot product of \( \mathbf{u} \) and \( \mathbf{v} \) was calculated as \(-5\sqrt{5} + 5\sqrt{5} + 2 = 2\).
Notice how the first two terms cancel each other out, leading to a simpler computation.

Magnitude of a Vector

The magnitude of a vector, also known as the vector's length, measures how long the vector is. It can be thought of like the distance from the origin to the point represented by the vector.

• For a vector \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k} \), its magnitude is given by \( \| \mathbf{v} \| = \sqrt{v_1^2 + v_2^2 + v_3^2} \).

In the provided solution, the vector \( \mathbf{v} \) has components \(-\sqrt{5} \), \( \sqrt{5} \), and \( 1 \) respectively.
Plugging these into the formula results in \( \| \mathbf{v} \| = \sqrt{5 + 5 + 1} = \sqrt{11} \).

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate square roots or radicals from the denominator of a fraction by multiplying both the numerator and the denominator by a suitable value. This process is often used in mathematics to make expressions easier to understand and work with.

• Given a fraction \( \frac{a}{\sqrt{b}} \), you rationalize the denominator by multiplying it by \( \frac{\sqrt{b}}{\sqrt{b}} \), which results in \( \frac{a\sqrt{b}}{b} \).

In our example, rationalizing the denominator of \( \frac{2}{\sqrt{11}} \) resulted in \( \frac{2\sqrt{11}}{11} \).
The process helped remove the radical from the denominator, yielding a more standardized form.

Vector Operations

Vector operations include various ways to manipulate vectors in different mathematical contexts. The basic operations involve addition, subtraction, scalar multiplication, and finding dot and cross products. These operations are essential tools in physics and engineering.

• Addition: Add corresponding components of two vectors.
• Subtraction: Subtract corresponding components of two vectors.
• Scalar Multiplication: Multiply each component of a vector by a scalar (a real number).
• Dot Product: As covered earlier, it's a scalar multiplication of two vectors.
• Cross Product: Produces a vector perpendicular to the plane containing the initial vectors (used primarily in three-dimensional space).

Understanding these operations is crucial for performing effective calculations and solving problems involving vectors.