Problem 26 - 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. \(\vec{u}=\langle 3,-1,2\rangle, \vec{v}=\langle 2,2,1\rangle\)

Step-by-Step Solution

Answer

The projection of \( \vec{u} \) onto \( \vec{v} \) is \( \langle \frac{4}{3}, \frac{4}{3}, \frac{2}{3} \rangle \).

Step 1: Understand the Formula

The orthogonal projection of vector \( \vec{u} \) onto vector \( \vec{v} \) is given by the formula \( \operatorname{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} \vec{v} \), where \( \cdot \) denotes the dot product.

Step 2: Calculate the Dot Product \(\vec{u} \cdot \vec{v}\)

Calculate the dot product \( \vec{u} \cdot \vec{v} = (3)(2) + (-1)(2) + (2)(1) = 6 - 2 + 2 = 6 \).

Step 3: Calculate the Dot Product \(\vec{v} \cdot \vec{v}\)

Calculate the dot product \( \vec{v} \cdot \vec{v} = (2)(2) + (2)(2) + (1)(1) = 4 + 4 + 1 = 9 \).

Step 4: Calculate the Projection Vector

Using the formula, substitute the values to find \( \operatorname{proj}_{\vec{v}} \vec{u} = \frac{6}{9} \vec{v} = \frac{2}{3} \langle 2, 2, 1 \rangle = \langle \frac{4}{3}, \frac{4}{3}, \frac{2}{3} \rangle \).

Step 5: Sketch the Vectors

Draw the vectors \( \vec{u} = \langle 3, -1, 2 \rangle \), \( \vec{v} = \langle 2, 2, 1 \rangle \), and \( \operatorname{proj}_{\vec{v}} \vec{u} = \langle \frac{4}{3}, \frac{4}{3}, \frac{2}{3} \rangle \) starting from the same initial point. \( \operatorname{proj}_{\vec{v}} \vec{u} \) will lie on the line of \( \vec{v} \).

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics focusing on vector fields and differential operators. It is particularly useful in physics and engineering.
Vectors in this context are quantities with both magnitude and direction, primarily used to represent physical quantities. They are often written in the format \(\langle x, y, z \rangle\) for three-dimensional spaces.
Vector calculus includes operations such as addition, subtraction, and projection of vectors. By understanding vector functions and using operations like the dot product, one can solve complex physical and geometrical problems.

• Vector operations allow for the modeling and analysis of real-world scenarios, like force fields and fluid dynamics.
• It relies heavily on concepts such as gradient, curl, and divergence that describe how vector fields change in space.

Understanding the basic operations in vector calculus, such as the orthogonal projection outlined in the exercise, is crucial for more advanced studies.

Dot Product

The dot product is a fundamental operation in vector calculus, which combines two vectors to produce a scalar value. It measures the magnitude of one vector in the direction of another.
This operation is essential as it not only provides a way to determine orthogonality but is also used to compute projections, like in the original exercise. The formula for dot product of vectors \(\vec{a} = \langle a_1, a_2, a_3 \rangle\) and \(\vec{b} = \langle b_1, b_2, b_3 \rangle\) is:
\[\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3\]
Key points about the dot product:

• If the dot product is zero, the vectors are orthogonal (perpendicular to each other).
• The dot product can help find angles between vectors, where \( \cos \theta = \frac{\vec{a} \cdot \vec{b}}{||\vec{a}|| ||\vec{b}||} \).
• It is crucial for calculating the orthogonal projection of a vector onto another vector.

In the exercise, the dot products \(\vec{u} \cdot \vec{v}\) and \(\vec{v} \cdot \vec{v}\) were calculated to find the projection of \(\vec{u}\) onto \(\vec{v}\).

Orthogonal Projection

Orthogonal projection is a technique used to "project" one vector onto another. It essentially decomposes a vector into a part parallel to another vector and a part perpendicular.
This is highly useful in both physics and computer graphics for simplifying complex problems and visual representations.
In practice, the orthogonal projection of a vector \(\vec{u}\) onto a vector \(\vec{v}\) can be calculated using:
\[\operatorname{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} \vec{v}\]
Key points include:

• Orthogonal projections compress the original vector onto the line created by another vector.
• The result remains on the line defined by \(\vec{v}\), effectively simplifying calculations.
• In certain contexts, this is used to reduce dimensions or eliminate potential errors from non-parallel components.

As applied in the exercise, it shows how one vector can be expressed in terms of another, aiding in problem-solving and optimization.

Three-Dimensional Vectors

Three-dimensional vectors are often used in spatial modeling, providing a way to describe and analyze physical space.
They are denoted as \(\langle x, y, z \rangle\) and are fundamental in depicting positions, directions, and forces in 3D space. Each component corresponds to an axis in the Cartesian coordinate system.
Understanding 3D vectors involves:

• Operations like vector addition and scaling, which allow for combining and adjusting vectors.
• Dot products and projections that enable complex calculations and analyses involving multiple vectors.
• Visualizing vectors as arrows in space, which can illuminate relationships between objects and phenomena.

In the exercise, both vectors \(\vec{u}\) and \(\vec{v}\) have three components, indicating how they operate within a 3D environment. The concepts taught by such exercises are crucial for fields like physics, engineering, and computer graphics.