Problem 76 - 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. $$ (\mathbf{u}-\mathbf{v}) \cdot(\mathbf{v}+\mathbf{w}) $$

Step-by-Step Solution

Answer

The result of the dot product is \(-16\).

Step 1: Find \( \mathbf{u} - \mathbf{v} \)

Subtract the vector \( \mathbf{v} = \langle -1, 1, 1 \rangle \) from the vector \( \mathbf{u} = \langle 1, -3, 2 \rangle \). \[ \mathbf{u} - \mathbf{v} = \langle 1 - (-1), -3 - 1, 2 - 1 \rangle = \langle 2, -4, 1 \rangle \]

Step 2: Find \( \mathbf{v} + \mathbf{w} \)

Add the vector \( \mathbf{v} = \langle -1, 1, 1 \rangle \) to the vector \( \mathbf{w} = \langle 2, 6, 9 \rangle \). \[ \mathbf{v} + \mathbf{w} = \langle -1 + 2, 1 + 6, 1 + 9 \rangle = \langle 1, 7, 10 \rangle \]

Step 3: Compute the Dot Product

Find the dot product of \( \mathbf{u} - \mathbf{v} = \langle 2, -4, 1 \rangle \) and \( \mathbf{v} + \mathbf{w} = \langle 1, 7, 10 \rangle \).Calculate it using the formula \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \):\[(2)(1) + (-4)(7) + (1)(10) = 2 - 28 + 10 = -16\]

Key Concepts

Understanding Vector Subtraction

Vector subtraction involves removing one vector from another. It's a simple operation that flips the direction of the vector being subtracted and then adds it to the first vector. Let's break it down:

Suppose we have two vectors, \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \). The subtraction \( \mathbf{a} - \mathbf{b} \) is done component-wise, resulting in the new vector \( \langle a_1 - b_1, a_2 - b_2, a_3 - b_3 \rangle \).

In our example, subtracting \( \mathbf{v} = \langle -1, 1, 1 \rangle \) from \( \mathbf{u} = \langle 1, -3, 2 \rangle \):

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

The result is \( \langle 2, -4, 1 \rangle \), a new vector pointing in a different direction and with a different magnitude.

The Essentials of Vector Addition

Vector addition is just as straightforward as subtraction. To add two vectors, simply join them tail-to-head and sum their corresponding components. If you have vectors \( \mathbf{c} = \langle c_1, c_2, c_3 \rangle \) and \( \mathbf{d} = \langle d_1, d_2, d_3 \rangle \), the result of \( \mathbf{c} + \mathbf{d} \) is \( \langle c_1 + d_1, c_2 + d_2, c_3 + d_3 \rangle \).

For the vectors in our exercise, add \( \mathbf{v} = \langle -1, 1, 1 \rangle \) and \( \mathbf{w} = \langle 2, 6, 9 \rangle \):

• First component: \(-1 + 2 = 1\)
• Second component: \(1 + 6 = 7\)
• Third component: \(1 + 9 = 10\)

Resulting in vector \( \langle 1, 7, 10 \rangle \). This operation is like finding a resultant vector that combines the influences of both original vectors.

Exploring Vector Operations with the Dot Product

Vector operations include various functions we can perform on vectors, with one of the most essential being the dot product. This operation tells us about the angle between two vectors, helping us understand their relationship.

The dot product of two vectors \( \mathbf{e} = \langle e_1, e_2, e_3 \rangle \) and \( \mathbf{f} = \langle f_1, f_2, f_3 \rangle \) is calculated as:\[\mathbf{e} \cdot \mathbf{f} = e_1f_1 + e_2f_2 + e_3f_3\]

Using the vectors from our problem, \( \mathbf{u} - \mathbf{v} = \langle 2, -4, 1 \rangle \) and \( \mathbf{v} + \mathbf{w} = \langle 1, 7, 10 \rangle \), we find:

• First components multiply: \((2)(1) = 2\)
• Second components multiply: \((-4)(7) = -28\)
• Third components multiply: \((1)(10) = 10\)

Add all these products together: \(2 + (-28) + 10\) to obtain the dot product: \(-16\). This negative value indicates that the vectors form an obtuse angle with each other.