Problem 24 - 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. \(\vec{u}=\langle-3,2\rangle, \vec{v}=\langle 2,3\rangle\)

Step-by-Step Solution

Answer

\(\text{proj}_{\vec{v}} \vec{u} = \langle 0, 0 \rangle\)

Step 1: Compute the dot products

\(\vec{u} \cdot \vec{v} = (-3)(2) + (2)(3) = -6 + 6 = 0\)
\(\vec{v} \cdot \vec{v} = 4 + 9 = 13\)

Step 2: Compute the projection

\(\text{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} \vec{v} = \frac{0}{13}\langle 2, 3\rangle = \langle 0, 0 \rangle\)

Step 3: Interpret the result

Since the projection is the zero vector, \(\vec{u}\) and \(\vec{v}\) are orthogonal (perpendicular). The projection of \(\vec{u}\) onto \(\vec{v}\) is the origin.

Key Concepts

Vectors

Vectors are mathematical objects that have both magnitude and direction, making them essential tools in various fields like physics and engineering. Think of a vector as an arrow in space. This arrow has a certain length (its magnitude) and points in a specific direction.

In terms of notation, a vector is typically represented as an ordered pair or triplet of numbers, depending on whether it is in 2D or 3D space, respectively. For instance, in a 2D space, a vector \( \vec{u} = \langle -3, 2 \rangle \) begins at the origin and extends to the point \( (-3, 2) \). Each component of the vector represents movement along the respective axis.

Why are vectors so useful?

• They allow you to represent quantities that have direction, like force and velocity.
• They make it possible to perform operations such as addition, subtraction, and scalar multiplication.
• Importantly, vectors enable powerful mathematical operations like projections and rotations in a coordinate space.

Understanding vectors allows us to comprehend more complex mathematical operations, like finding the angle between vectors or projecting one vector onto another.

Dot Product

The dot product is a way to multiply two vectors to result in a scalar. It is a crucial part of understanding vectors and their projections.

The formula for the dot product of two vectors \( \vec{u} = \langle a_1, a_2 \rangle \) and \( \vec{v} = \langle b_1, b_2 \rangle \) in 2D is: \(\vec{u} \cdot \vec{v} = a_1 b_1 + a_2 b_2\)
This operation simplifies calculations when dealing with vectors and allows us to measure how much one vector extends in the direction of another.

The dot product has some fascinating properties:

• It is commutative, meaning \( \vec{u} \cdot \vec{v} = \vec{v} \cdot \vec{u} \).
• If the dot product is zero, the vectors are orthogonal (at a right angle to each other).
• It relates directly to the cosine of the angle between two vectors. If you know the magnitudes of the vectors and the angle \( \theta \) between them, the dot product can also be computed as \( \vec{u} \cdot \vec{v} = || \vec{u} || || \vec{v} || \cos(\theta) \).

Hence, the dot product is not just about arithmetic but also about understanding the relationship between vectors.

Projection Formula

The projection of one vector onto another is used to determine how much of one vector extends in the direction of another. This is especially useful in applications where figuring out one vector's component in the direction of another is necessary, such as in physics when decomposing forces.

The formula for the projection of vector \( \vec{u} \) onto vector \( \vec{v} \) is:\[ \operatorname{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} \vec{v}\]
This formula scales the direction of \( \vec{v} \) by the proportion of \( \vec{u} \'s \) presence in \( \vec{v} \. \)Let's break this down:

• The numerator \( \vec{u} \cdot \vec{v} \) gives us the dot product, telling us about the directional overlap between \( \vec{u} \) and \( \vec{v} \).
• The denominator \( \vec{v} \cdot \vec{v} \) normalizes the calculation, dividing by the length squared of \( \vec{v} \) to scale properly.
• The result, \( \operatorname{proj}_{\vec{v}} \vec{u} \, \) is a vector that shows how far \( \vec{u} \) extends along \( \vec{v} \).

When applied, this projection provides insight into the geometry and relationships of vectors in a space, allowing for a deeper understanding of vector operations and their real-world applications.