Problem 20 - Precalculus : Building Concepts and Connections

Question

Calculate projev. Then decompose \(\mathbf{v}\) into \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\) $$\mathbf{v}=\langle 1,2\rangle, \mathbf{w}=\langle-3,3\rangle$$

Step-by-Step Solution

Answer

The projection of \( \mathbf{v} \) on \( \mathbf{w} \) is \( \mathbf{v}_{1} = \langle -0.5,0.5 \rangle \) and vector \( \mathbf{v}_{2} \) orthogonal to \( \mathbf{w} \) is \( \langle 1.5, 1.5 \rangle \).

Step 1: Calculate the Projection of \( \mathbf{v} \) on \( \mathbf{w} \)

The projection of \( \mathbf{v} \) on \( \mathbf{w} \) can be found using the formula \[ projev = (\frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{w}\|^2})\mathbf{w} \] Substituting the given vectors \( \mathbf{v} \) and \( \mathbf{w} \) into this formula yields \[ projev =(\frac{\langle1, 2 \rangle \cdot \langle -3, 3 \rangle}{\| \langle -3,3 \rangle \|^2}) \langle -3, 3 \rangle \] = (\frac{-3+6}{9+9}) \langle -3,3 \rangle = (\frac{3}{18}) \langle -3,3 \rangle = \langle -0.5,0.5 \rangle. \] So, \( \mathbf{v}_{1} \) is \(\langle -0.5,0.5 \rangle.\)

Step 2: Calculate \( \mathbf{v}_{2} \) orthogonal to \( \mathbf{w} \)

After finding \( \mathbf{v}_{1} \), which is the projection of \( \mathbf{v} \) on \( \mathbf{w} \), we can find \( \mathbf{v}_{2} \) by subtracting the projection vector from \( \mathbf{v} \). Using the formula \( \mathbf{v}_{2} = \mathbf{v} - projev = \langle 1, 2 \rangle - \langle -0.5, 0.5 \rangle = \langle 1.5, 1.5 \rangle \).

Key Concepts

Projection of a Vector

The projection of one vector onto another is like casting a shadow onto a line. Imagine shining a light on vector \(\mathbf{v}\) in the direction of another vector \(\mathbf{w}\); the shadow on \(\mathbf{w}\) is the projection of \(\mathbf{v}\) onto \(\mathbf{w}\).
This concept is crucial in various applications, such as physics and engineering, where it's necessary to break down forces into components.To calculate the projection of vector \(\mathbf{v}\) on \(\mathbf{w}\), we use the formula:

• Find the dot product, \(\mathbf{v} \cdot \mathbf{w}\), by multiplying corresponding components and adding them together.
• Calculate the magnitude squared of \(\mathbf{w}\), which involves squaring each component of \(\mathbf{w}\) and summing them up.

Use these results to find the coefficient that scales \(\mathbf{w}\), giving the shadow—this is the projection vector. The projection formula is \(projev = \left( \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{w}\|^2} \right) \mathbf{w}\).
In our exercise, the projection is \(\langle -0.5,0.5 \rangle\).

Orthogonal Vectors

Vectors are said to be orthogonal if they are perpendicular to each other. This is an important concept because orthogonal vectors have a dot product of zero, indicating no shared direction.
Orthogonality is used in many scenarios, including optimization and least squares problems to ensure independence between vectors. To find a component of vector \(\mathbf{v}\) that is orthogonal to another vector \(\mathbf{w}\), you subtract the projection of \(\mathbf{v}\) on \(\mathbf{w}\) from the initial vector \(\mathbf{v}\). Using the formula

• \(\mathbf{v}_{2} = \mathbf{v} - projev\), where \(projev\) is the previously calculated projection.

In our example, subtracting \(\langle -0.5,0.5 \rangle\) from \(\langle 1,2 \rangle\) gives a resulting orthogonal vector \(\langle 1.5, 1.5 \rangle\).

Vector Operations

Vectors can be manipulated in many ways, akin to numbers in mathematics, and vector operations include addition, subtraction, dot product, and scalar multiplication. Each operation serves different functions and can be applied using specific rules.
These operations enable transformations such as rotation, scaling, and translation in space.
For example, subtraction or addition involves moving a vector according to another's direction and magnitude, while the dot product reveals the cosine of the angle between them, showing similarity in direction.Key operations include:

• Addition and Subtraction: Add or subtract corresponding components.
• Dot Product: Multiply corresponding components and sum the results.
• Scalar Multiplication: Multiply each vector component by a scalar.

These operations underpin the decomposition of vectors, helping to isolate components aligned or orthogonal to another vector, as seen in breaking \(\mathbf{v}\) into \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\).