Problem 68 - 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. $$ 4(\mathbf{u}+2 \mathbf{w})-6 \mathbf{v} $$

Step-by-Step Solution

Answer

The vector is \( \langle 26, 30, 74 \rangle \).

Step 1: Calculate the Scalar Multiplication with Vector w

First, multiply vector \( \mathbf{w} = \langle 2,6,9 \rangle \) by the scalar 2. This results in \( 2 \cdot \mathbf{w} = \langle 2\times2, 6\times2, 9\times2 \rangle = \langle 4, 12, 18 \rangle \).

Step 2: Add Vector u and Result of Step 1

Now, add vector \( \mathbf{u} = \langle 1, -3, 2 \rangle \) with the result from step 1 \( \langle 4, 12, 18 \rangle \). This gives us \( \mathbf{u} + 2\mathbf{w} = \langle 1+4, -3+12, 2+18 \rangle = \langle 5, 9, 20 \rangle \).

Step 3: Multiply the Result by Scalar 4

Multiply the result from Step 2 by scalar 4: \( 4 \cdot \langle 5, 9, 20 \rangle = \langle 4\times5, 4\times9, 4\times20 \rangle = \langle 20, 36, 80 \rangle \).

Step 4: Calculate Scalar Multiplication with Vector v

Multiply vector \( \mathbf{v} = \langle -1, 1, 1 \rangle \) by scalar 6: \( 6\mathbf{v} = \langle 6\times(-1), 6\times1, 6\times1 \rangle = \langle -6, 6, 6 \rangle \).

Step 5: Subtract Step 4 Result from Step 3 Result

Subtract the result from Step 4 from the result of Step 3: \( \langle 20, 36, 80 \rangle - \langle -6, 6, 6 \rangle = \langle 20 + 6, 36 - 6, 80 - 6 \rangle = \langle 26, 30, 74 \rangle \).

Key Concepts

Scalar Multiplication

Scalar multiplication is one of the basic operations in vector algebra. It involves multiplying a vector by a scalar (a plain number). This operation scales the vector by the given amount. If the scalar is positive, the direction of the vector remains the same, while a negative scalar reverses it.

For example, if we have a vector \( \mathbf{w} = \langle 2, 6, 9 \rangle \) and a scalar 2, then scalar multiplication gives us \( 2 \mathbf{w} = \langle 2\times2, 6\times2, 9\times2 \rangle = \langle 4, 12, 18 \rangle \).

• Each component of the vector is multiplied by the scalar.
• The resultant vector is parallel to the original vector.
• Scalar multiplication can enlarge or reduce the size of the vector, based on the scalar's absolute value.

Understanding scalar multiplication is crucial, as it is often combined with other operations like vector addition and subtraction.

Vector Addition

Vector addition is another fundamental operation where two or more vectors are added together to form a new vector. This is done by adding corresponding components of the vectors.

For instance, take vectors \( \mathbf{u} = \langle 1, -3, 2 \rangle \) and \( \mathbf{w} = \langle 4, 12, 18 \rangle \). Adding these vectors component-wise gives:

• First component: \(1 + 4 = 5\)
• Second component: \(-3 + 12 = 9\)
• Third component: \(2 + 18 = 20\)

This results in the vector \( \mathbf{u} + 2 \mathbf{w} = \langle 5, 9, 20 \rangle \).

Vector addition follows the commutative property (\( \mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a} \)) and associative property ((\( \mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c}) \)). It's important for combining different vector operations.

Vector Subtraction

Vector subtraction is similar to vector addition but involves subtracting the components of one vector from the corresponding components of another vector. This is essentially adding a vector and the negative of another vector.

For example, if we have vectors \( \langle 20, 36, 80 \rangle \) and \( \langle -6, 6, 6 \rangle \), subtraction is executed as:

• First component: \(20 + 6 = 26\)
• Second component: \(36 - 6 = 30\)
• Third component: \(80 - 6 = 74\)

This produces the resultant vector \( \langle 26, 30, 74 \rangle \).

Just like addition, vector subtraction is crucial when you need to determine a vector's change or difference from another. It's a simple flip of sign and addition, making it easy to perform.

Three-Dimensional Vectors

Three-dimensional vectors are vectors that have three components, representing the x, y, and z axes in three-dimensional space. They are written in the form \( \langle x, y, z \rangle \). These vectors are significant in physics, engineering, and computer graphics because they effectively model real-world scenarios.

Take the vectors from the exercise, like \( \mathbf{u} = \langle 1, -3, 2 \rangle \), \( \mathbf{v} = \langle -1, 1, 1 \rangle \), and \( \mathbf{w} = \langle 2, 6, 9 \rangle \). Each of these denotes a position or a direction in space.

• The x-component affects movement along the horizontal axis.
• The y-component affects vertical movement.
• The z-component influences depth, showing movement forward or backward.

Understanding three-dimensional vectors helps in visualizing and solving problems for objects moving through space, making concepts in dynamics and spatial analysis much clearer.