Problem 29 - Precalculus with Limits

Question

Finding the Magnitude of a Vector In Exercises \(25-30\) , use the dot product to find the magnitude of u. $$\mathbf{u}=6 \mathbf{j}$$

Step-by-Step Solution

Answer

The magnitude of the vector \( \mathbf{u} = 6 \mathbf{j}\) is 6.

Step 1: Understanding the formula for magnitude of a vector

The magnitude (or length) of a vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \) in 3D space is given by the square root of the sum of the squares of its components. In equation form, this is \( |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2}\) . However, note that for our vector \( \mathbf{u} = 6 \mathbf{j} \), we only have a \(\mathbf{j}\) component of 6 and no \(\mathbf{i}\) or \( \mathbf{k}\) components.

Step 2: Applying the formula

Therefore, to calculate the magnitude of our vector \( \mathbf{u} \), we can apply the formula for magnitude of a vector by inserting our known values: \( | \mathbf{u} | = \sqrt{6^2} = \sqrt{36} = 6\).

Key Concepts

Dot Product

The dot product is a method of multiplying two vectors together. It results in a single scalar quantity rather than another vector. For two vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \), the dot product is given by: \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).
This operation is particularly helpful in physics and engineering, where you often need to find angles between vectors or project one vector onto another. In the context of finding the magnitude, the dot product can simplify the calculation process when dealing with self-dot products like \( \mathbf{u} \cdot \mathbf{u} \). In the example provided, \( \mathbf{u} \cdot \mathbf{u} = 6 \mathbf{j} \cdot 6 \mathbf{j} = 36 \), leading to the magnitude: \( \sqrt{36} = 6 \).
Use the dot product to identify orthogonality between vectors as well. If two non-zero vectors' dot product is zero, they are orthogonal.

Vector Components

Every vector in three-dimensional space consists of components along the standard unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \). These components specify how much of the vector lies along each axis: x, y, and z axes respectively.
For example, a vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \) has three components:

• \( a \) along the x-axis (\( \mathbf{i} \))
• \( b \) along the y-axis (\( \mathbf{j} \))
• \( c \) along the z-axis (\( \mathbf{k} \))

In the exercise, the vector \( \mathbf{u} = 6 \mathbf{j} \) showcases only a y-component while other components are zero.
This simplifies the calculation of the magnitude, as you only need to consider the non-zero component. However, understanding vector components is essential for resolving complex three-dimensional movement and forces.

Three-Dimensional Space

Three-dimensional space is the environment in which most real-world phenomena occur, involving three axes—x, y, and z. Vitally, vectors in this space become instrumental in describing directions and magnitudes (e.g., velocity, force). This space is key for disciplines ranging from physics to computer graphics.
The structure of 3D vectors, \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \), advocates a deeper understanding of three-dimensional geometry. This understanding allows for robust modeling of interactions and positions within this space.
When learning about vectors in 3D, it's useful to visualize how each component of the vector influences its direction and overall vector magnitude, as demonstrated in exercises where vectors inhabit only one axis extensively, like \( \mathbf{u} = 6 \mathbf{j} \). Vector visualization can be an excellent tool for comprehension, enabling you to see vectors in action.

Vector Notation

Vector notation is the concise system used to express vectors and their components efficiently. In three-dimensional space, vectors are typically expressed as a sum of unit vectors.
For instance, the vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \) employs unit vectors \( \mathbf{i}, \mathbf{j}, \text{and} \ \mathbf{k} \) to indicate direction along the x, y, and z axes respectively.
In the exercise example, \( \mathbf{u} = 6 \mathbf{j} \) is displayed in vector notation, showing a clean representation of its y-component and absence of x and z-components.
Mastering vector notation is crucial as it simplifies formula usage and problem solving, allowing accurate and quick identification of vector components, which in turn streamlines calculations like finding the magnitude.