Problem 34 - Algebra and Trigonometry

Question

Find \(\operatorname{proj}_{w} \mathbf{v}\) . Then decompose v into two vectors, \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to w and \(\mathbf{v}_{2}\) is orthogonal to w. $$ \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}, \quad \mathbf{w}=2 \mathbf{i}+\mathbf{j} $$

Step-by-Step Solution

Answer

The projection of v onto w is \(proj_{w} \mathbf{v} =1.6 \mathbf{i}+0.8 \mathbf{j}\). The component of v that is parallel to w, v1, is \(v1 = 1.6 \mathbf{i}+0.8 \mathbf{j}\) and the component of v that is orthogonal to w, v2, is \(v2 = 1.4 \mathbf{i} - 2.8 \mathbf{j}\)

Step 1: Calculate Dot Product

Firstly, the dot product of vectors v and w is calculated as \(v \cdot w = (3*2)+(-2*1) = 4\)

Step 2: Calculate Magnitude of Vector w

Next, calculate the magnitude of w squared, which is \(||w||^2 = (2^2) + (1^2) = 5\)

Step 3: Calculate projection of v onto w

Now, calculate the projection of v onto w, which we will represent as \(proj_{w} \mathbf{v}\). For vector projection, the formula is \((v \cdot w)/(||w||^2)*w\). Substitute the values calculated in previous steps to get \(proj_{w} \mathbf{v}\) = [(4/5)*2]*i + [(4/5)*1]*j. Simplify to get \(proj_{w} \mathbf{v} =1.6 \mathbf{i}+0.8 \mathbf{j}\)

Step 4: Calculate the component of v parallel to w

The component of v parallel to w is just the projection of v onto w. Therefore, v1 = \(proj_{w} \mathbf{v}\). Thus, v1 = 1.6 \mathbf{i}+0.8 \mathbf{j}

Step 5: Calculate the component of v orthogonal to w

The component of v orthogonal to w, v2, can be found by subtracting v1 from v. Therefore, v2 = v - v1. Thus, v2 = (3 \mathbf{i} - 2 \mathbf{j}) - (1.6 \mathbf{i} + 0.8 \mathbf{j}). Simplify to get v2 = 1.4 \mathbf{i} - 2.8 \mathbf{j}

Key Concepts

Dot Product

Understanding the dot product, also known as the scalar product, is central to many vector operations. It's a way to multiply two vectors together and get a scalar (a single number) as the result, rather than another vector. To calculate it, you take the corresponding components of each vector, multiply them, and then sum those products. For two-dimensional vectors \textbf{A} and \textbf{B} with components \textbf{A} = (a1, a2) and \textbf{B} = (b1, b2), the dot product is calculated as: \[ A \text{dot} B=a1 \times b1 + a2 \times b2 \]This product has applications in determining the angle between vectors and in projecting one vector onto another.

Vector Magnitude

The magnitude of a vector is the length or size of the vector and is a measure of the distance from its starting point to its end point. For a vector \textbf{v} with components (v1, v2), the magnitude (denoted as \( ||\textbf{v}|| \) ) is found by taking the square root of the sum of the squares of its components. Mathematically, we express this as: \[ ||\textbf{v}|| = \text{square root of} (v1^2 + v2^2) \]Knowing the magnitude is essential for various vector operations, including normalization, where a vector is scaled to have a magnitude of one without changing its direction.

Orthogonal Vectors

Vectors are orthogonal to each other if they meet at a right angle (90 degrees). An important property of orthogonal vectors is that their dot product equals zero: if vectors \textbf{A} and \textbf{B} are orthogonal, then \( A \text{dot} B = 0 \). This makes checking for orthogonality quite straightforward. It's a crucial concept for understanding vector decomposition, where one vector is broken down into components parallel and orthogonal to another.

Vector Decomposition

Vector decomposition involves breaking down a vector into several parts or components. This is particularly useful when dealing with forces in physics or simplifying complex vector calculations. The idea is to represent a vector as the sum of two or more vectors, typically one component being parallel to another vector and the other being orthogonal. In mathematical terms, if \textbf{v} is our original vector and \textbf{w} is the vector we’re projecting onto, then \textbf{v} can be decomposed into \( \textbf{v}_{1} \) and \( \textbf{v}_{2} \), where \( \textbf{v}_{1} \) is parallel to \( \textbf{w} \) and \( \textbf{v}_{2} \) is orthogonal to \( \textbf{w} \).