Question

In Exercises 21-26, 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. \(\vec{u}=\langle 1,2\rangle, \vec{v}=\langle-1,3\rangle\)

Short answer

The projection is \( \langle -\frac{1}{2}, \frac{3}{2} \rangle \).

Step-by-step solution

Understand the Formula

The orthogonal projection of \( \vec{u} \) onto \( \vec{v} \) is given by the formula: \( \operatorname{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} \vec{v} \). This formula calculates how much of the vector \( \vec{u} \) goes in the direction of \( \vec{v} \).

Compute the Dot Product \(\vec{u} \cdot \vec{v}\)

Calculate the dot product of \( \vec{u} = \langle 1, 2 \rangle \) and \( \vec{v} = \langle -1, 3 \rangle \): \( \vec{u} \cdot \vec{v} = 1 \times (-1) + 2 \times 3 = -1 + 6 = 5 \).

Compute the Dot Product \(\vec{v} \cdot \vec{v}\)

Calculate the dot product of \( \vec{v} = \langle -1, 3 \rangle \) with itself: \( \vec{v} \cdot \vec{v} = (-1)^2 + 3^2 = 1 + 9 = 10 \).

Calculate the Projection

Use the values from Steps 2 and 3 in the projection formula: \[ \operatorname{proj}_{\vec{v}} \vec{u} = \frac{5}{10} \vec{v} = \frac{1}{2} \langle -1, 3 \rangle = \langle -\frac{1}{2}, \frac{3}{2} \rangle. \]

Interpret the Result

The vector \( \langle -\frac{1}{2}, \frac{3}{2} \rangle \) is the orthogonal projection of \( \vec{u} \) onto \( \vec{v} \). It represents the shadow of \( \vec{u} \) on the line defined by \( \vec{v} \).

Key concepts

Dot Product

The dot product is a fundamental operation in vector mathematics. It is used to find the relationship between two vectors. When you calculate the dot product of two vectors \( \vec{a} = \langle a_1, a_2 \rangle \) and \( \vec{b} = \langle b_1, b_2 \rangle \), the formula is \( \vec{a} \cdot \vec{b} = a_1 \times b_1 + a_2 \times b_2 \). This calculation results in a single number, not a vector.

The dot product gives us a measure of how much one vector goes in the direction of another. If the dot product is positive, the vectors are pointing in roughly the same direction. A negative dot product means they are going in opposite directions, while zero indicates they are perpendicular.

In the example from the step-by-step solution, the dot product \( \vec{u} \cdot \vec{v} \) helps in finding the orthogonal projection by showing how much of \( \vec{u} \) aligns with \( \vec{v} \). After computing, we see that \( \vec{u} \cdot \vec{v} = 5 \), meaning some component of \( \vec{u} \) is aligned with \( \vec{v} \).

Understanding the dot product is essential for mastering concepts like projections, angles between vectors, and can even help in determining vector orthogonality.

Orthogonal Projection

Orthogonal projection is a useful technique in vector mathematics. It helps in finding a vector that represents the shadow or footprint of one vector on another. When we project vector \( \vec{u} \) onto vector \( \vec{v} \), we want to determine the portion of \( \vec{u} \) that aligns with \( \vec{v} \).

The formula to calculate this projection is:
\[ \operatorname{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} \vec{v} \]
Here, the numerator \( \vec{u} \cdot \vec{v} \) gives the component of \( \vec{u} \) in the direction of \( \vec{v} \), while the denominator \( \vec{v} \cdot \vec{v} \) normalizes this component with respect to the length of \( \vec{v} \). This results in a vector that is parallel to \( \vec{v} \).

From the solution steps, we find the orthogonal projection to be \( \langle -\frac{1}{2}, \frac{3}{2} \rangle \). This vector is what you would get if you dropped a perpendicular from the endpoint of \( \vec{u} \) down to the line extending along \( \vec{v} \).

Using orthogonal projections is particularly helpful in various applications like physics, engineering, and computer graphics, where resolving forces or motion into components is crucial.

Vector Mathematics

Vector mathematics is the study of vectors, which are mathematical constructs representing quantities with both magnitude and direction. Understanding vectors is crucial in various fields such as physics, computer science, and engineering.

Vectors are often represented in a coordinate form like \( \langle x, y \rangle \) in two dimensions or \( \langle x, y, z \rangle \) in three dimensions. They possess unique properties:

• Addition: Vectors can be added together using their components. If \( \vec{a} = \langle a_1, a_2 \rangle \) and \( \vec{b} = \langle b_1, b_2 \rangle \), then \( \vec{a} + \vec{b} = \langle a_1 + b_1, a_2 + b_2 \rangle \).
• Scalar Multiplication: Vectors can be multiplied by scalars (numbers) to stretch or shrink their length while maintaining direction.
• Magnitude: The length of a vector, given by \( \| \vec{v} \| = \sqrt{x^2 + y^2} \) in two dimensions.

In problem-solving, vectors allow for the decomposition of problems into manageable parts. They're used in defining transformations, modeling physical phenomena, and solving geometric problems.

From the exercise, understanding vector operations like the dot product and orthogonal projection is vital. These concepts allow us to manipulate vectors in a way that solves real-world problems efficiently and accurately. In the context of the exercise, recognizing how vectors project onto each other showcases their practical application in analyzing directional components.