Problem 65 - Precalculus with Calculus Previews

Question

In Problems \(65-76, \mathbf{u}=\langle 1,-3,2\rangle, \mathbf{v}=\langle-1,1,1\rangle\), and \(\mathbf{w}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. $$ \mathbf{u}+(\mathbf{v}+\mathbf{w}) $$

Step-by-Step Solution

Answer

\( \mathbf{u} + (\mathbf{v} + \mathbf{w}) = \langle 2, 4, 12 \rangle \).

Step 1: Calculate the sum of vectors \( \mathbf{v} \) and \( \mathbf{w} \)

First, you need to add the vectors \( \mathbf{v} = \langle -1, 1, 1 \rangle \) and \( \mathbf{w} = \langle 2, 6, 9 \rangle \). The addition of vectors is done by adding their corresponding components: \( (-1) + 2 = 1 \), \( 1 + 6 = 7 \), and \( 1 + 9 = 10 \). Thus, \( \mathbf{v} + \mathbf{w} = \langle 1, 7, 10 \rangle \).

Step 2: Add vector \( \mathbf{u} \) to the result from Step 1

Now, we add the vector \( \mathbf{u} = \langle 1, -3, 2 \rangle \) to the result obtained from Step 1, which is \( \langle 1, 7, 10 \rangle \). Again, add the corresponding components: \( 1 + 1 = 2 \), \( -3 + 7 = 4 \), and \( 2 + 10 = 12 \). Therefore, \( \mathbf{u} + (\mathbf{v} + \mathbf{w}) = \langle 2, 4, 12 \rangle \).

Key Concepts

Vectors

Vectors are essential elements in mathematics and physics, representing quantities with both magnitude and direction. Think of them as arrows on a graph, where each arrow shows not only the size or amount of something but also the direction it is pointing.
It's like a much-needed road map if you want to find your way through problems involving movement, force, or even other complex mathematical operations.

• Each vector is typically represented in a notation like \( \mathbf{a} = \langle x, y, z \rangle \), where \( x \), \( y \), and \( z \) are called components.
• The arrows or lines in this vector notation visually tell us in which direction the vector is acting and by how much on each axis.

From movement pathways in physics to data points in statistics, vectors help us understand and solve a wide range of problems.
They can add, subtract, or be scaled as necessary, all thanks to their flexibility and multifaceted nature.

Components of a Vector

The components of a vector are like ingredients in a recipe—they determine the vector's final shape and effectiveness. When handling vectors like \( \mathbf{u} = \langle 1, -3, 2 \rangle \), these numbers (1, -3, and 2) tell us how far the vector moves along the x, y, and z axes, respectively.
Understanding this concept is crucial when performing operations like vector addition, as it involves dealing with each component individually.
Here's a simple breakdown of how components work in vector mathematics:

• **X-component**: Moves left or right along the horizontal axis.
• **Y-component**: Moves up or down along the vertical axis.
• **Z-component**: Used in 3D to shift forward or backward depth-wise.

When adding vectors, you focus only on adding these components separately—a process that's like stacking building blocks to form a more complex structure.
This clear act of separateness ensures the math mirrors real-world phenomena accurately, maintaining the integrity of direction and magnitude.

Scalar

In the world of vectors, scalars are their simpler siblings. Unlike vectors that carry both direction and size, scalars are single values representing only magnitude, like speed or temperature.
One of the most distinct features of scalars is their inability to tell you which direction something is going—they offer no directional information.

• **Magnitude Only**: Scalars only have size, meaning they can't be used for directional analysis.
• **Representation**: Often denoted as plain numbers or symbols, like \( \, 5 \, \) for size, without angled brackets.

When you involve scalars in calculations with vectors, they act as multipliers or dividers, changing the size of a vector without affecting its direction.
By keeping an eye on this distinction, it's easier to distinguish when to use vectors or scalars for effective analysis in varied mathematical scenarios.