/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 Find $$\mathbf{a}+\mathbf{b}, 2 ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 17

Find $$\mathbf{a}+\mathbf{b}, 2 \mathbf{a}+3 \mathbf{b},|\mathbf{a}|, \text { and }|\mathbf{a} \mathbf{b}|$$ $$\mathbf{a}=\langle 5,-12\rangle, \quad \mathbf{b}=\langle- 3,-6\rangle$$

Short Answer

Expert verified
\(\mathbf{a} + \mathbf{b} = \langle 2, -18 \rangle\), \(2\mathbf{a} + 3\mathbf{b} = \langle 1, -42 \rangle\), \(|\mathbf{a}| = 13\)."

Step by step solution

01

Adding Vectors \(\mathbf{a} + \mathbf{b}\)

To find \(\mathbf{a} + \mathbf{b}\), add the corresponding components of each vector. Given \(\mathbf{a} = \langle 5, -12 \rangle\) and \(\mathbf{b} = \langle -3, -6 \rangle\), add the x-components and y-components:\[\mathbf{a} + \mathbf{b} = \langle 5 + (-3), -12 + (-6) \rangle = \langle 2, -18 \rangle\]
02

Linear Combination \(2\mathbf{a} + 3\mathbf{b}\)

To find \(2\mathbf{a} + 3\mathbf{b}\), first multiply each vector by their scalar and then add the results. Compute:\[2\mathbf{a} = 2 \times \langle 5, -12 \rangle = \langle 10, -24 \rangle\]\[3\mathbf{b} = 3 \times \langle -3, -6 \rangle = \langle -9, -18 \rangle\]Add these results:\[2\mathbf{a} + 3\mathbf{b} = \langle 10, -24 \rangle + \langle -9, -18 \rangle = \langle 1, -42 \rangle\]
03

Calculate the Magnitude \(|\mathbf{a}|\)

The magnitude of a vector \(\mathbf{a} = \langle x, y \rangle\) is given by \(|\mathbf{a}| = \sqrt{x^2 + y^2}\). For \(\mathbf{a} = \langle 5, -12 \rangle\),\[|\mathbf{a}| = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13\]
04

Checking Question on \(|\mathbf{a} \mathbf{b}|\)

There is no standard vector operation denoted as \(|\mathbf{a} \mathbf{b}|\). If it means the magnitude of the dot product of \(\mathbf{a}\) and \(\mathbf{b}\), it should be calculated as \(\mathbf{a} \cdot \mathbf{b}\). The dot product is:\[\mathbf{a} \cdot \mathbf{b} = (5)(-3) + (-12)(-6) = -15 + 72 = 57\]However, since a vector multiplication type isn't specified, we should assume no answer is needed for this section.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Vector Addition
Vector addition is a fundamental operation in vector mathematics that combines two vectors to create a resultant vector. Picture two arrows on a graph representing vectors. Adding them involves shifting one vector so that its tail resides at the head of the other, then drawing a new arrow from the start of the fixed vector to the endpoint of the second.
  • To add vectors, sum up their corresponding components. For instance, for vectors \( \mathbf{a} = \langle 5, -12 \rangle \) and \( \mathbf{b} = \langle -3, -6 \rangle \), add \( 5 + (-3) = 2 \) and \( -12 + (-6) = -18 \).
  • The resulting vector is \( \langle 2, -18 \rangle \). This new vector illustrates the additive synthesis of \( \mathbf{a} \) and \( \mathbf{b} \).
Vector addition is visually intuitive and essential for solving many physics and engineering problems.
Scalar Multiplication
Scalar multiplication involves stretching or compressing a vector by a scalar value. Think of changing the length of a string without altering its orientation.
  • Each vector component gets multiplied by the scalar: for \( 2 \mathbf{a} \), multiply each part of \( \mathbf{a} = \langle 5, -12 \rangle \) by 2, yielding \( \langle 10, -24 \rangle \).
  • Similarly, for \( 3 \mathbf{b} \), with \( \mathbf{b} = \langle -3, -6 \rangle \), scale each component by 3, resulting in \( \langle -9, -18 \rangle \).
After scaling, combine the vectors using addition: \( 2\mathbf{a} + 3\mathbf{b} = \langle 10, -24 \rangle + \langle -9, -18 \rangle = \langle 1, -42 \rangle \). This operation changes the vector's size without altering its direction.
Magnitude of a Vector
The magnitude of a vector represents its length, analogous to the hypotenuse in a right triangle.To find the magnitude of vector \( \mathbf{a} = \langle x, y \rangle \), apply the Pythagorean theorem:
  • Compute \( |\mathbf{a}| = \sqrt{x^2 + y^2} \).
  • For \( \mathbf{a} = \langle 5, -12 \rangle \), this equates to \( \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \).
Understanding vector magnitude is crucial for determining the vector's scale and orientation in space.
Dot Product
The dot product of two vectors yields a scalar result and is valuable for finding angles between vectors or determining orthogonality.To calculate the dot product, multiply corresponding vector components and sum them:
  • For vectors \( \mathbf{a} = \langle 5, -12 \rangle \) and \( \mathbf{b} = \langle -3, -6 \rangle \), compute \( \mathbf{a} \cdot \mathbf{b} = (5)(-3) + (-12)(-6) = -15 + 72 = 57 \).
  • This operation's result, 57, reveals the degree of parallelism or angle between the vectors.
Dot products are key in physics to determine work done or in computer graphics for lighting calculations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use intercepts to help sketch the plane. \(6 x+5 y-3 z=15\)

Find equations of the planes that are parallel to the plane \(x+2 y-2 z=1\) and two units away from it.

Sketch the region bounded by the paraboloids \(z=x^{2}+y^{2}\) and \(z=2-x^{2}-y^{2}\)

Show that if the point \((a, b, c)\) lies on the hyperbolic paraboloid \(z=y^{2}-x^{2},\) then the lines with parametric equations \(x=a+t, y=b+t, z=c+2(b-a) t\) and \(x=a+t\) \(y=b-t, z=c-2(b+a) t\) both lie entirely on this paraboloid. (This shows that the hyperbolic paraboloid is what is called a ruled surface; that is, it can be generated by the motion of a straight line. In fact, this exercise shows that through each point on the hyperbolic paraboloid there are two generating lines. The only other quadric surfaces that are ruled surfaces are cylinders, cones, and hyperboloids of one sheet.)

Find the distance between the given parallel planes. \(2 x-3 y+z=4, \quad 4 x-6 y+2 z=3\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.