Problem 23 - APEX Calculus

Question

Vectors \(\vec{u}\) and \(\vec{v}\) are given. Find \(\operatorname{proj}_{\vec{v}} \vec{u}\) the orthogonal projection of \(\vec{u}\) onto \(\vec{v},\) and sketch all three vectors with the same initial point. \(\vec{u}=\langle-3,2\rangle, \vec{v}=\langle 1,1\rangle\)

Step-by-Step Solution

Answer

The projection of \( \vec{u} \) onto \( \vec{v} \) is \( \left\langle -\frac{1}{2}, -\frac{1}{2} \right\rangle \).

Step 1: Understand the Formula for Projection

The orthogonal projection of a vector \( \vec{u} \) onto another vector \( \vec{v} \) is given by the formula \( \operatorname{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} \cdot \vec{v} \). This allows us to express \( \operatorname{proj}_{\vec{v}} \vec{u} \) as a scalar multiple of the vector \( \vec{v} \).

Step 2: Compute the Dot Products

Calculate the dot product \( \vec{u} \cdot \vec{v} = (-3)(1) + (2)(1) = -3 + 2 = -1 \). Then calculate \( \vec{v} \cdot \vec{v} = (1)(1) + (1)(1) = 1 + 1 = 2 \).

Step 3: Calculate the Projection

Plug the dot products into the projection formula: \( \operatorname{proj}_{\vec{v}} \vec{u} = \frac{-1}{2} \cdot \langle 1, 1 \rangle = \langle -\frac{1}{2}, -\frac{1}{2} \rangle \). The projection vector is \( \left\langle -\frac{1}{2}, -\frac{1}{2} \right\rangle \).

Step 4: Sketch the Vectors

Draw the vector \( \vec{u} = \langle -3, 2 \rangle \) starting from the origin. Then draw the vector \( \vec{v} = \langle 1, 1 \rangle \), which should also start from the origin. Finally, draw \( \operatorname{proj}_{\vec{v}} \vec{u} = \left\langle -\frac{1}{2}, -\frac{1}{2} \right\rangle \) from the origin, and ensure it lies along \( \vec{v} \). Confirm that \( \operatorname{proj}_{\vec{v}} \vec{u} \) is perpendicular to the line orthogonal to \( \vec{v} \) and intersects \( \vec{u} \).

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics, crucial for calculations in vector projection. It is a way to multiply two vectors to obtain a scalar, or a single number. When you have two vectors, say \( \vec{u} = \langle a_1, b_1, c_1 \rangle \) and \( \vec{v} = \langle a_2, b_2, c_2 \rangle \) in 3-dimensional space, their dot product is calculated as:

• \( \vec{u} \cdot \vec{v} = a_1 \cdot a_2 + b_1 \cdot b_2 + c_1 \cdot c_2 \)

This helps determine the angle between the two vectors. If the dot product equals zero, the vectors are orthogonal, meaning they are at a 90-degree angle to each other.
For the problem at hand, we found the dot product \( \vec{u} \cdot \vec{v} = -1 \) and \( \vec{v} \cdot \vec{v} = 2 \). This calculation laid the groundwork for finding the orthogonal projection.

Orthogonal Projection

Orthogonal projection is the process of projecting a vector onto another vector to find its "shadow."
Imagine the sun casting shadows; the shadow of an object on the ground mimics the object's presence but lies in the direction of the light source. Similarly, the orthogonal projection reveals part of one vector that is parallel to another.The formula for orthogonal projection is:

• \( \operatorname{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} \cdot \vec{v} \)

This formula gives us a new vector, \( \operatorname{proj}_{\vec{v}} \vec{u} \), which lies along \( \vec{v} \) and is closest to \( \vec{u} \).
Therefore, it is a part of \( \vec{u} \) that lies entirely in the direction of \( \vec{v} \), ignoring any part that's perpendicular.
In our example, this projection ended up as \( \langle -\frac{1}{2}, -\frac{1}{2} \rangle \).
It affirms that the blue and red vectors create a right angle in a traditionally colored stereoscopic image.

Euclidean Vector Spaces

Euclidean vector spaces are the playground for vector operations, characterized by vectors that follow the familiar rules of flat plane geometry.
These spaces allow us to visualize complex mathematical operations like projection using Cartesian coordinates (the x, y, and sometimes z coordinates you know from graphing in math class). Elements in these spaces, or vectors, are depicted as arrows that have direction and magnitude.
Key properties of Euclidean vector spaces include:

• They are equipped with operations such as vector addition and scalar multiplication.
• They have a notion of measure for vector lengths and angles between vectors.
• In these spaces, the concept of orthogonality and projection can be visually interpreted.

Since our vectors \( \vec{u} \) and \( \vec{v} \) are in a two-dimensional Euclidean space, all these operationsoccur on a flat plane.
This space is perfect for seeing how vectors interact with each other geometrically,helping ensure the projection and vector relationships make visual sense.