Problem 33 - Precalculus Mathematics for Calculus

Question

Calculate proj, u. (b) Resolve u into \(u_{1}\) and \(u_{2}\), where \(\mathbf{u}_{1}\) is parallel to \(\mathbf{v}\) and \(\mathbf{u}_{2}\) is orthogonal to \(\mathbf{v}\). $$\mathbf{u}=\langle 2,9\rangle, \quad \mathbf{v}=\langle- 3,4\rangle$$

Step-by-Step Solution

Answer

proj of u onto v is \( \langle -3.6, 4.8 \rangle \); \( \mathbf{u}_1 = \langle -3.6, 4.8 \rangle \), \( \mathbf{u}_2 = \langle 5.6, 4.2 \rangle \).

Step 1: Calculate Projection of u onto v

First, we need to calculate the projection of \( \mathbf{u} \) onto \( \mathbf{v} \), which is denoted as \( \text{proj}_{\mathbf{v}} \mathbf{u} \). The formula is: \[ \text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v} \]Calculate \( \mathbf{u} \cdot \mathbf{v} = 2(-3) + 9(4) = -6 + 36 = 30 \).Calculate \( \mathbf{v} \cdot \mathbf{v} = (-3)^2 + 4^2 = 9 + 16 = 25 \).Thus, \[ \text{proj}_{\mathbf{v}} \mathbf{u} = \frac{30}{25} \langle -3, 4 \rangle = \langle -3.6, 4.8 \rangle \]

Step 2: Resolve u into u1 and u2

Now, separate \( \mathbf{u} \) into components \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \), where \( \mathbf{u}_1 \) is parallel to \( \mathbf{v} \) and \( \mathbf{u}_2 \) is orthogonal to \( \mathbf{v} \).We found \( \mathbf{u}_1 = \text{proj}_{\mathbf{v}} \mathbf{u} = \langle -3.6, 4.8 \rangle \).Now, compute \( \mathbf{u}_2 \) using \( \mathbf{u}_2 = \mathbf{u} - \mathbf{u}_1 \).Therefore, \( \mathbf{u}_2 = \langle 2, 9 \rangle - \langle -3.6, 4.8 \rangle = \langle 2 + 3.6, 9 - 4.8 \rangle = \langle 5.6, 4.2 \rangle \).

Key Concepts

Dot Product

To understand the projection of a vector, knowing about the dot product is essential. The dot product of two vectors, sometimes called the scalar product, provides a way to multiply vectors to get a scalar value. Two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) have a dot product calculated as:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \)

This operation helps in measuring how aligned two vectors are:

• When the dot product is positive, the vectors point in a similar direction.
• When it's negative, they point in opposite directions.
• A dot product of zero suggests orthogonality (being at right angles).

This formula is crucial in many applications, such as finding vector projections or understanding vector orientation.
In this specific example, we calculated the dot product of \( \mathbf{u} = \langle 2, 9 \rangle \) and \( \mathbf{v} = \langle -3, 4 \rangle \) as 30, indicating some alignment between these vectors, as they are not orthogonal.

Orthogonal Vectors

Orthogonal vectors have an interesting property: they are perpendicular to each other. In two-dimensional or three-dimensional space, this means they form a 90-degree angle. Orthogonality is identified when the dot product of two vectors is zero, indicating no direct overlap or alignment with each other.
Consider two vectors \( \mathbf{a} \) and \( \mathbf{b} \). If \( \mathbf{a} \cdot \mathbf{b} = 0 \), then these vectors are orthogonal. This means any component of one vector doesn’t affect the other in terms of direction.
From the exercise, once we determined \( \mathbf{u}_1 \), the parallel component of \( \mathbf{u} \), the orthogonal component \( \mathbf{u}_2 \) was easily found by subtracting \( \mathbf{u}_1 \) from \( \mathbf{u} \). This is how \( \mathbf{u}_2 = \langle 5.6, 4.2 \rangle \) was calculated, ensuring it’s orthogonal to \( \mathbf{v} \).

Parallel Vectors

Parallel vectors share the same direction or are exactly opposite; one is a scalar multiple of the other. This means that one vector could simply be a scaled version of another.
In the context of projection, determining a vector's parallel component helps depict how much of a vector aligns with another. The formula used here is

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

This calculates the part of vector \( \mathbf{u} \) that runs parallel to vector \( \mathbf{v} \).
For the example, the projection \( \text{proj}_{\mathbf{v}} \mathbf{u} \) resulted in \( \langle -3.6, 4.8 \rangle \), which became \( \mathbf{u}_1 \). This means \( \mathbf{u}_1 \) lies along the direction of \( \mathbf{v} \), portraying the full extent of \( \mathbf{u} \) that is aligned with \( \mathbf{v} \). This component is crucial in multiple vector-related applications, such as physics and computer graphics, to determine influence or directional flow.