Problem 29 - APEX Calculus

Question

Vectors \(\vec{u}\) and \(\vec{v}\) are given. Write \(\vec{u}\) as the sum of two vectors, one of which is parallel to \(\vec{v}\) and one of which is perpendicular to \(\vec{v}\). Note: these are the same pairs of vectors as found in Exercises 21-26. \(\vec{u}=\langle-3,2\rangle, \vec{v}=\langle 1,1\rangle\)

Step-by-Step Solution

Answer

\(\vec{u}\) is the sum of \(\langle -\frac{1}{2}, -\frac{1}{2}\rangle\) and \(\langle -\frac{5}{2}, \frac{5}{2}\rangle\).

Step 1: Find the Projection of u onto v

To find the vector parallel to \( \vec{v} \), we need to calculate the projection of \( \vec{u} \) onto \( \vec{v} \). The formula for the projection of \( \vec{u} \) onto \( \vec{v} \) is given by:\[ \text{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{\vec{v} \cdot \vec{v}} \vec{v} \]First, calculate the dot product \( \vec{u} \cdot \vec{v} = (-3 \times 1) + (2 \times 1) = -3 + 2 = -1 \).Then, calculate \( \vec{v} \cdot \vec{v} = (1 \times 1) + (1 \times 1) = 1 + 1 = 2 \).The projection is:\[ \text{proj}_{\vec{v}} \vec{u} = \frac{-1}{2} \langle 1, 1 \rangle = \langle -\frac{1}{2}, -\frac{1}{2} \rangle \].

Step 2: Find the Vector Perpendicular to v

The vector perpendicular to \( \vec{v} \) is found by subtracting the projection from \( \vec{u} \). That is:\[ \vec{u}_{\perp} = \vec{u} - \text{proj}_{\vec{v}} \vec{u} \]Using the values from Step 1, we have:\[ \vec{u}_{\perp} = \langle -3, 2 \rangle - \langle -\frac{1}{2}, -\frac{1}{2} \rangle = \langle -3 + \frac{1}{2}, 2 + \frac{1}{2} \rangle = \left\langle -\frac{6}{2} + \frac{1}{2}, \frac{4}{2} + \frac{1}{2} \right\rangle \]Simplifying:\[ \vec{u}_{\perp} = \left\langle -\frac{5}{2}, \frac{5}{2} \right\rangle \].

Step 3: Write u as the Sum of the Two Vectors

Combine the vectors from Steps 1 and 2. The vector \( \vec{u} \) can be expressed as the sum of the parallel and perpendicular components:\[ \vec{u} = \text{proj}_{\vec{v}} \vec{u} + \vec{u}_{\perp} \]Plug in the values:\[ \vec{u} = \langle -\frac{1}{2}, -\frac{1}{2} \rangle + \langle -\frac{5}{2}, \frac{5}{2} \rangle \]This confirms that \( \vec{u} = \langle -3, 2 \rangle \) as expected.

Key Concepts

Vector Projection

Vector projection helps identify how much of one vector lies along the direction of another vector. This concept is essential in breaking down vectors into components, as used in many fields like physics and engineering.
The vector projection of vector \( \vec{u} \) onto vector \( \vec{v} \) is denoted as \( \text{proj}_{\vec{v}} \vec{u} \). It is given by the formula:

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

Here’s what happens:

• The dot product \( \vec{u} \cdot \vec{v} \) is calculated as \(-3 + 2 = -1\).
• The dot product \( \vec{v} \cdot \vec{v} \) splits into \(1 + 1 = 2\).
• Use these in the projection formula to get \( \langle -\frac{1}{2}, -\frac{1}{2} \rangle \).

This projected vector is the 'shadow' of \( \vec{u} \) that's pointing in the same direction as \( \vec{v} \).
Understanding vector projection is key to analyzing vector-related problems.

Dot Product

The dot product is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number.
In this exercise, the dot product is crucial in determining both the projection of one vector onto another and understanding the relationship between these vectors.
For two vectors \( \vec{u} = \langle u_1, u_2 \rangle \) and \( \vec{v} = \langle v_1, v_2 \rangle \), the dot product is calculated as:

• \( \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 \)

What happens here:

• For \( \vec{u} \cdot \vec{v} \), it equals \(-3 + 2 = -1\).
• For \( \vec{v} \cdot \vec{v} \), it equals \(1 + 1 = 2\).

The dot product helps measure how closely two vectors align. If the dot product is zero, vectors are perpendicular. Here, the dot product tells us about the projection, assisting in vector decomposition.

Parallel Vectors

When talking about parallel vectors, we're discussing vectors that lie in the same direction or exactly opposite directions.
They can have different magnitudes but must align along the same line in space.
In vector projection, the resulting projection is a parallel component of the original vector. This exercise's vector \( \langle -\frac{1}{2}, -\frac{1}{2} \rangle \) represents the component of \( \vec{u} \) that is parallel to \( \vec{v} \).
Characteristics of parallel vectors include:

• Identical or precisely opposite direction.
• Magnitude may differ.

Understanding parallel vectors can help visualize vector relationships and simplify complex vector operations.

Perpendicular Vectors

Perpendicular vectors form an angle of 90° (right angle) with each other.
This makes them extremely useful when decomposing vectors, allowing us to explore components independently without interference.
In the context of the exercise, the perpendicular component is found by calculating the difference between the original vector and the projection (parallel component):

• \( \vec{u}_{\perp} = \langle -3, 2 \rangle - \langle -\frac{1}{2}, -\frac{1}{2} \rangle \)

This results in \( \langle -\frac{5}{2}, \frac{5}{2} \rangle \), which is perpendicular to \( \vec{v} \).
Properties of perpendicular vectors include:

• Their dot product is zero.
• They allow independent analysis of scenarios, especially in physics problems.

Recognizing perpendicular vectors aids in clear division of vector spaces and efficient problem solving.