Problem 39 - Advanced Engineering Mathematics

Question

In Problems \(39-42\), find proj \(_{\mathrm{b}}\). \(\mathbf{a}=-5 \mathbf{i}+5 \mathbf{j}, \mathbf{b}=-3 \mathbf{i}+4 \mathbf{j}\)

Step-by-Step Solution

Answer

The projection of \(\mathbf{a}\) onto \(\mathbf{b}\) is \(-\frac{21}{5} \mathbf{i} + \frac{28}{5} \mathbf{j}\)."

Step 1: Formula for Projection

To find the projection of vector \(\mathbf{a}\) onto vector \(\mathbf{b}\), we use the formula: \(\text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}\). This formula involves calculating the dot product of the vectors and then scaling vector \(\mathbf{b}\).

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

The dot product \(\mathbf{a} \cdot \mathbf{b}\) is calculated by multiplying corresponding components of \(\mathbf{a}\) and \(\mathbf{b}\) and summing them up. So, \(\mathbf{a} \cdot \mathbf{b} = (-5)(-3) + (5)(4) = 15 + 20 = 35\).

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

The dot product \(\mathbf{b} \cdot \mathbf{b}\) is simply the sum of the squares of its components. Thus, \(\mathbf{b} \cdot \mathbf{b} = (-3)^2 + (4)^2 = 9 + 16 = 25\).

Step 4: Calculate Projection of \(\mathbf{a}\) onto \(\mathbf{b}\)

Now, substitute the values obtained from Steps 2 and 3 into the projection formula: \(\text{proj}_{\mathbf{b}} \mathbf{a} = \frac{35}{25} \mathbf{b} = \frac{7}{5}\mathbf{b}\).

Step 5: Scale Vector \(\mathbf{b}\)

Scale vector \(\mathbf{b}= -3 \mathbf{i} + 4 \mathbf{j}\) by \(\frac{7}{5}\): \(\text{proj}_{\mathbf{b}} \mathbf{a} = \frac{7}{5}(-3 \mathbf{i}) + \frac{7}{5}(4 \mathbf{j}) = -\frac{21}{5} \mathbf{i} + \frac{28}{5} \mathbf{j}\).

Key Concepts

Understanding the Dot Product

When working with vectors, understanding the dot product is essential. The dot product of two vectors is a scalar, computed by multiplying corresponding components and then summing the results. It is mathematically expressed as: \[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n \]For our vectors \(\mathbf{a}= -5\mathbf{i} + 5\mathbf{j}\) and \(\mathbf{b}= -3\mathbf{i} + 4\mathbf{j}\), the dot product is: \[(-5)(-3) + (5)(4) = 15 + 20 = 35\]This value will play a crucial role in further calculations, as it is part of the projection formula and essentially measures how much one vector goes in the direction of the other.

Scaling Vectors

Vector scaling involves multiplying a vector by a scalar. This process changes the vector's magnitude but retains its direction.Here's how it works for vector \( \mathbf{b} \):- Original vector: \( \mathbf{b} = -3 \mathbf{i} + 4 \mathbf{j} \)- Scale by \( \frac{7}{5} \): - \[ \frac{7}{5}(-3) = -\frac{21}{5} \] - \[ \frac{7}{5}(4) = \frac{28}{5} \]The resulting scaled vector is \(-\frac{21}{5} \mathbf{i} + \frac{28}{5} \mathbf{j}\). This scaled vector is actually the projection of vector \( \mathbf{a} \) onto \( \mathbf{b} \). Scaling is thus a fundamental concept for finding projections.

Projection Formula Explained

The projection of one vector onto another is determined using the projection formula. This formula uses the dot product and scaling to find out how much of one vector runs in the direction of another. The formula is:\[\text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}\]Let's break it down:- The numerator \( \mathbf{a} \cdot \mathbf{b} \) represents how much \( \mathbf{a} \) aligns with \( \mathbf{b} \), which we've calculated as 35.- The denominator \( \mathbf{b} \cdot \mathbf{b} \) is actually the magnitude squared of \( \mathbf{b} \) and here it comes out as 25.- We then divide these values and scale \( \mathbf{b} \), resulting in: \[\text{proj}_{\mathbf{b}} \mathbf{a} = \frac{35}{25} \mathbf{b} = \frac{7}{5} \mathbf{b}\]This formula transforms all these calculations into a single, intuitive expression for projection.

Advanced Mathematics in Vector Projections

In advanced mathematics, vector projections are crucial in many fields such as physics, engineering, and computer graphics. Understanding this concept helps solve complex problems in vector calculus and linear algebra. Here are a few key explorations: - **Orthogonal Projections**: Learning projections also involve understanding when they are perpendicular, known as orthogonal. It's a concept deeply rooted in geometry and spacing. - **Applications in Physics**: Consider forces in mechanics where projecting forces along certain directions can simplify calculations. - **Computer Graphics**: Here, projections play a significant role in rendering 3D scenes onto 2D screens using various projection techniques. Grasp these advanced topics by solidifying your understanding of the fundamental laws governing vectors and projections. They form the groundwork for more intricate theories and applications.