Problem 16 - Precalculus Mathematics for Calculus

Question

Find the vectors \(\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v},\) and \(3 \mathbf{u}-\frac{1}{2} \mathbf{v}\) $$\mathbf{u}=\langle 0,1,-3\rangle, \quad v=\langle 4,2,0\rangle$$

Step-by-Step Solution

Answer

\(\mathbf{u}+\mathbf{v}=\langle 4,3,-3\rangle\), \(\mathbf{u}-\mathbf{v}=\langle -4,-1,-3\rangle\), \(3\mathbf{u}-\frac{1}{2}\mathbf{v}=\langle -2,2,-9\rangle\)."

Step 1: Define the vectors

Start by defining the given vectors. We have \( \mathbf{u} = \langle 0, 1, -3 \rangle \) and \( \mathbf{v} = \langle 4, 2, 0 \rangle \).

Step 2: Calculate \(\mathbf{u} + \mathbf{v}\)

We add the corresponding components of vectors \(\mathbf{u}\) and \(\mathbf{v}\). \[ \mathbf{u} + \mathbf{v} = \langle 0+4, 1+2, -3+0 \rangle = \langle 4, 3, -3 \rangle \]

Step 3: Calculate \(\mathbf{u} - \mathbf{v}\)

Next, we subtract the corresponding components of \(\mathbf{v}\) from \(\mathbf{u}\).\[ \mathbf{u} - \mathbf{v} = \langle 0-4, 1-2, -3-0 \rangle = \langle -4, -1, -3 \rangle \]

Step 4: Calculate \(3 \mathbf{u}\) and \(-\frac{1}{2} \mathbf{v}\) separately

First, calculate \(3 \mathbf{u}\) by multiplying each component of \(\mathbf{u}\) by 3.\[ 3 \mathbf{u} = \langle 3 \times 0, 3 \times 1, 3 \times (-3) \rangle = \langle 0, 3, -9 \rangle \]Then, calculate \(-\frac{1}{2} \mathbf{v}\) by multiplying each component of \(\mathbf{v}\) by \(-\frac{1}{2}\).\[ -\frac{1}{2} \mathbf{v} = \langle -\frac{1}{2} \times 4, -\frac{1}{2} \times 2, -\frac{1}{2} \times 0 \rangle = \langle -2, -1, 0 \rangle \]

Step 5: Calculate \(3 \mathbf{u} - \frac{1}{2} \mathbf{v}\)

Finally, add the vectors obtained in Step 4.\[ 3 \mathbf{u} - \frac{1}{2} \mathbf{v} = \langle 0, 3, -9 \rangle + \langle -2, -1, 0 \rangle = \langle 0 + (-2), 3 + (-1), -9 + 0 \rangle = \langle -2, 2, -9 \rangle \]

Key Concepts

Vector Subtraction

Subtraction of vectors involves taking each component of one vector and subtracting the corresponding component of another vector. Imagine you have two vectors, for instance, \( \mathbf{u} = \langle 0, 1, -3 \rangle \) and \( \mathbf{v} = \langle 4, 2, 0 \rangle \). To find \( \mathbf{u} - \mathbf{v} \), you subtract each component of \( \mathbf{v} \) from each component of \( \mathbf{u} \) like so:

• First component: \( 0 - 4 = -4 \)
• Second component: \( 1 - 2 = -1 \)
• Third component: \( -3 - 0 = -3 \)

As a result, the vector \( \mathbf{u} - \mathbf{v} \) is \( \langle -4, -1, -3 \rangle \).
Vector subtraction is a simple yet powerful tool for finding differences between directions and magnitudes in space.

Scalar Multiplication

Scalar multiplication involves multiplying each component of a vector by a scalar (a constant number). Let's imagine vector \( \mathbf{u} = \langle 0, 1, -3 \rangle \) and we want to find \( 3 \mathbf{u} \), which means multiplying the vector by the scalar 3. Here's how it works:

• Multiply the first component by 3: \( 3 \times 0 = 0 \)
• Multiply the second component by 3: \( 3 \times 1 = 3 \)
• Multiply the third component by 3: \( 3 \times -3 = -9 \)

So, \( 3 \mathbf{u} = \langle 0, 3, -9 \rangle \). Scalar multiplication is useful for scaling vectors, changing their length but not their direction. An interesting twist is multiplying by negative scalars, which reverses the vector's direction.

Three-Dimensional Vectors

Three-dimensional vectors are all about navigating through space in three directions: x, y, and z. Each component of a 3D vector represents a dimension in this space. Vectors like \( \mathbf{u} = \langle 0, 1, -3 \rangle \) live in a world where each of those numbers
represents movement along one of these axes.

• The x-component tells you how far right or left to go
• The y-component describes how far up or down
• The z-component, often the trickiest, indicates movement backward or forward out of our familiar two-dimensional view

These vectors are crucial in fields ranging from physics, where they describe forces and velocities, to computer graphics, where they control camera positions and lighting in 3D scenes.
Three-dimensional vectors give us the mathematical tools to represent real-world spatial relationships in calculations and visualizations.