Problem 41 - Advanced Engineering Mathematics

Question

In Problems \(39-42\), find proj \(_{\mathrm{b}}\). \(\mathbf{a}=-\mathbf{i}-2 \mathbf{j}+7 \mathbf{k}, \mathbf{b}=6 \mathbf{i}-3 \mathbf{j}-2 \mathbf{k}\)

Step-by-Step Solution

Answer

The projection of \( \mathbf{a} \) onto \( \mathbf{b} \) is \(-\frac{12}{7} \mathbf{i} + \frac{6}{7} \mathbf{j} + \frac{4}{7} \mathbf{k}\).

Step 1: Understand the Projection Formula

The vector projection of \( \mathbf{a} \) onto \( \mathbf{b} \), denoted as \( \text{proj}_{\mathbf{b}} \mathbf{a} \), is given by the formula \[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b} \]. To solve the given problem, we'll first compute \( \mathbf{a} \cdot \mathbf{b} \) and \( \mathbf{b} \cdot \mathbf{b} \).

Step 2: Calculate the Dot Product \( \mathbf{a} \cdot \mathbf{b} \)

The dot product is calculated by multiplying corresponding components of the vectors and adding them together: \( \mathbf{a} \cdot \mathbf{b} = (-1)(6) + (-2)(-3) + (7)(-2) = -6 + 6 - 14 = -14 \).

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

Calculate the dot product of \( \mathbf{b} \) with itself: \( \mathbf{b} \cdot \mathbf{b} = (6)(6) + (-3)(-3) + (-2)(-2) = 36 + 9 + 4 = 49 \).

Step 4: Use the Projection Formula

Substitute the calculated values into the projection formula: \[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{-14}{49} \mathbf{b} \]. Simplify the scalar multiplication: \( \frac{-14}{49} = -\frac{2}{7} \).

Step 5: Find the Projection Vector

Multiply the vector \( \mathbf{b} \) by the scalar \(-\frac{2}{7}\) to find the projection: \[ \text{proj}_{\mathbf{b}} \mathbf{a} = -\frac{2}{7} (6 \mathbf{i} - 3 \mathbf{j} - 2 \mathbf{k}) = -\frac{12}{7} \mathbf{i} + \frac{6}{7} \mathbf{j} + \frac{4}{7} \mathbf{k} \].

Key Concepts

dot product

The dot product, sometimes referred to as the scalar product, is a key operation in vector algebra. It combines two vectors to produce a scalar, or single number. To compute the dot product of two vectors, you multiply corresponding components and then sum these results. Consider two vectors \( \mathbf{u} = u_1 \mathbf{i} + u_2 \mathbf{j} + u_3 \mathbf{k} \) and \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k} \). The dot product is calculated as:

• \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \)

In practice, the dot product measure helps reveal the angle between vectors and is crucial in determining parallelism and orthogonality.
For example, in our problem, calculating the dot product \( \mathbf{a} \cdot \mathbf{b} = -14 \) involves:

• Multiplying each component: \((-1)(6), (-2)(-3), (7)(-2)\)
• Summing results: \(-6 + 6 - 14\)

The dot product can tell us a lot about vector interactions.

vector algebra

Vector algebra is the branch of mathematics dealing with quantities that have both magnitude and direction, commonly represented as vectors. Key operations in vector algebra include dot product, cross product, and vector addition. Each operation serves different purposes:

• Dot product provides a scalar, useful for projections and angles.
• Cross product yields a vector, perpendicular to original ones.
• Vector addition results in a vector found by "adding" the directions.

Vector algebra is used extensively in physics and engineering to describe phenomena such as forces, motion, and fields. In solving the problem of finding vector projections, vector algebra helps manage how one vector stretches or shadows another.
Understanding vector algebra is crucial as it helps in breaking down complex problems into simpler components. This allows for easier manipulation and solution finding.

scalar multiplication

Scalar multiplication is the process of multiplying a vector by a scalar (a real number). This operation stretches or shrinks the vector by altering its magnitude while keeping its direction the same.For instance, scaling \( \mathbf{v} = 6 \mathbf{i} - 3 \mathbf{j} - 2 \mathbf{k} \) by a scalar of \(-\frac{2}{7} \) involves:

• Multiplying each vector component by \(-\frac{2}{7} \):
• \(6 \times -\frac{2}{7} = -\frac{12}{7}\)
• \(-3 \times -\frac{2}{7} = \frac{6}{7}\)
• \(-2 \times -\frac{2}{7} = \frac{4}{7}\)

This yields the new vector: \(-\frac{12}{7} \mathbf{i} + \frac{6}{7} \mathbf{j} + \frac{4}{7} \mathbf{k} \).
Scalar multiplication is essential in vector projection as it determines how much of one vector lies along another's direction, which is critical when solving projection problems like the one presented.