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Chapter 7: Problem 85

Find the dot product of the vectors \((-2,6)\) and \(\langle 3,5\rangle\)

Short Answer

Expert verified
The dot product is 24.

Step by step solution

01

Understand the dot product formula

The dot product of two vectors \((-2, 6)\) and \(\textbf{b}=\begin{pmatrix} 3 \ 5 \) is found using the formula: \(\textbf{a} \bullet \textbf{b} = a_1 \times b_1 + a_2 \times b_2\), where \(a_1, a_2\) and \(b_1, b_2\) are the components of vectors \(\textbf{a}\) and \(\textbf{b}\), respectively.
02

Identify vector components

For the vectors \(\textbf{a}=(-2, 6)\) and \(\textbf{b}=\begin{pmatrix} 3 \ 5 \), the components are \(a_1=-2, a_2=6, b_1=3,\) and \(b_2=5\).
03

Multiply corresponding components

Next, multiply the first components of both vectors: \(-2 \times 3 = -6\).
04

Multiply the second components

Multiply the second components: \(6 \times 5 = 30\).
05

Add the products

Add the results of the multiplications: \(-6 + 30 = 24\).
06

State the dot product

The dot product of the vectors \((-2, 6)\) and \(\begin{pmatrix} 3 \ 5 \) is 24.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Components
Vectors are fundamental in mathematics and physics, representing quantities that have both magnitude and direction. To better understand and work with vectors, it is crucial to break them down into their components.
A vector in a two-dimensional plane can be written as \([a_1, a_2]\) or \(\).\

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Most popular questions from this chapter

Determine whether each pair of vectors is parallel, perpendicular, or neither. $$\langle 5,3\rangle,\langle 2,5\rangle$$

Determine whether each pair of vectors is parallel, perpendicular, or neither. $$\langle 2,3\rangle,\langle 8,12\rangle$$

Find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\) for each pair of complex numbers, using trigonometric form. Write the answer in the form \(a+b i\). $$z_{1}=2-6 i, z_{2}=-3-2 i$$

Given that \(\mathbf{A}=\langle 3,1\rangle\) and \(\mathbf{B}=\langle- 2,3\rangle,\) find the magnitude and direction angle for each of the following vectors. $$\mathbf{A}-\mathbf{B}$$

Write each complex number in the form \(a+b i\). $$4\left(\cos 180^{\circ}+i \sin 180^{\circ}\right)$$
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