Problem 20 - Algebra and Trigonometry

Question

19-22 Find the component of \(\mathbf{u}\) along \(\mathbf{v}\). $$\mathbf{u}=\langle- 3,5), \quad \mathbf{v}=\langle 1 / \sqrt{2}, 1 / \sqrt{2}\rangle$$

Step-by-Step Solution

Answer

The component of \( \mathbf{u} \) along \( \mathbf{v} \) is \( \sqrt{2} \).

Step 1: Understand the Concept

The component of vector \( \mathbf{u} \) along vector \( \mathbf{v} \) represents how much of \( \mathbf{u} \) is in the direction of \( \mathbf{v} \). This is calculated using the dot product and the magnitude of the direction vector.

Step 2: Calculate the Dot Product

The dot product of two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is given by: \( a_1b_1 + a_2b_2 \). For our vectors, we have: \[ \mathbf{u} \cdot \mathbf{v} = (-3) \times \frac{1}{\sqrt{2}} + 5 \times \frac{1}{\sqrt{2}} \] Simplifying gives: \[ \mathbf{u} \cdot \mathbf{v} = \frac{-3}{\sqrt{2}} + \frac{5}{\sqrt{2}} = \frac{2}{\sqrt{2}} \] Which simplifies to: \[ \mathbf{u} \cdot \mathbf{v} = \sqrt{2} \]

Step 3: Calculate the Magnitude of \( \mathbf{v} \)

The magnitude of a vector \( \mathbf{b} = \langle b_1, b_2 \rangle \) is \( \sqrt{b_1^2 + b_2^2} \). Thus: \[ ||\mathbf{v}|| = \sqrt{\left( \frac{1}{\sqrt{2}} \right)^2 + \left( \frac{1}{\sqrt{2}} \right)^2} \] Simplifying gives: \[ ||\mathbf{v}|| = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1 \]

Step 4: Compute the Component

The component of \( \mathbf{u} \) along \( \mathbf{v} \) is given by the formula: \[ \text{component} = \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{v}||} \] Substitute the values: \[ \text{component} = \frac{\sqrt{2}}{1} = \sqrt{2} \]

Key Concepts

Dot Product

The dot product is a fundamental operation used in vector calculus often to find the relationship between two vectors. It essentially calculates the "overlap" of one vector in the direction of another. For two 2-dimensional vectors, say \(\mathbf{a} = \langle a_1, a_2 \rangle\) and \(\mathbf{b} = \langle b_1, b_2 \rangle\), the dot product is computed as \(a_1b_1 + a_2b_2\). This operation effectively measures how much of one vector points in the direction of the other.

• It's a scalar, not a vector.
• If two vectors are perpendicular, their dot product is zero.
• It's essential in calculating angles between vectors and projections.

In the provided exercise, we found the dot product between \(\mathbf{u}\) and \(\mathbf{v}\) to be \(\sqrt{2}\). This value was key in determining how much of \(\mathbf{u}\) projects onto \(\mathbf{v}\). Understanding this operation helps students grasp how vectors interact in a shared space.

Magnitude of a Vector

The magnitude of a vector, often referred to as its length, indicates how long the vector is in a spatial sense. For a vector \(\mathbf{b} = \langle b_1, b_2 \rangle\), its magnitude is calculated using the formula \(\sqrt{b_1^2 + b_2^2}\). It's the equivalent of determining distance from the origin to the point represented by the vector in a coordinate system.

• Magnitude gives the exact "size" of the vector, ignoring direction.
• It embodies the concept of how far a point (end of the vector) is from the origin.
• Always a non-negative value and only zero if the vector is a zero vector.

In the example, the magnitude of \(\mathbf{v}\) was calculated as \(1\), a crucial step because it allows us to express vectors in unit form, making other calculations, such as projections, straightforward. Expressing vectors as unit vectors simplifies many vector operations in both theoretical and practical applications.

Vector Direction

The direction of a vector tells us the path that the vector points towards, distinguished from its magnitude which indicates how far. Knowing the direction is vital when resolving vectors into components or when analyzing motion and forces in physics. Especially in cases of projections, understanding direction allows us to find how much a vector contributes along another vector's path.

• Direction is usually expressed in terms of angles or coordinates.
• It is as meaningful as magnitude in defining vector behavior and interaction.
• Can indicate orientations such as north, east, up, down, etc., depending on the application.

In the exercise, \(\mathbf{v}\) is expressed in a direction ideal for calculating components — since \(\mathbf{v}\) is a unit vector, the operations become much more straightforward. Understanding vector direction allows us to predict outcomes in various dynamics, making it indispensable in both geometric and physical analyses.