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Chapter 13: Problem 47

For two-dimensional vectors \(\vec{a}\) and \(\vec{b},\) if \(\|\vec{a}\|=2\) and \(\|\vec{b}\|=4,\) find \(\|\vec{a}+\vec{b}\|\) for the given \(\vec{a} \cdot \vec{b}\) $$\vec{a} \cdot \vec{b}=-8$$

Short Answer

Expert verified
The magnitude of \(\vec{a} + \vec{b}\) is 2.

Step by step solution

01

Understand the Magnitude Formula

To find the magnitude of the sum of two vectors, we use the formula \[\|\vec{a} + \vec{b}\| = \sqrt{\|\vec{a}\|^2 + \|\vec{b}\|^2 + 2\vec{a} \cdot \vec{b}}.\]
02

Calculate \(\|\vec{a}\|^2\) and \(\|\vec{b}\|^2\)

\[\|\vec{a}\|^2 = 2^2 = 4.\] \[\|\vec{b}\|^2 = 4^2 = 16.\] We have calculated the squares of the magnitudes.
03

Substitute Values into the Magnitude Formula

Insert the known values into the magnitude formula: \[\|\vec{a} + \vec{b}\| = \sqrt{4 + 16 + 2(-8)}.\]
04

Simplify Inside the Square Root

Simplify the expression under the square root: \[4 + 16 + 2(-8) = 20 - 16 = 4.\]
05

Find the Magnitude

Calculate the square root to find the magnitude: \[\|\vec{a} + \vec{b}\| = \sqrt{4} = 2.\]

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Key Concepts

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

Magnitude of a Vector
The magnitude of a vector, often called its length or norm, measures how long the vector is, irrespective of its direction. In a two-dimensional space, if you have a vector \( \vec{v} = (x, y) \), the magnitude is represented by \( \|\vec{v}\| \). Calculating the magnitude involves the Pythagorean theorem.
  • You square the component values \((x^2 + y^2)\).
  • Add them together.
  • Then take the square root to find the overall length: \( \|\vec{v}\| = \sqrt{x^2 + y^2} \).
So when the magnitudes of vectors \( \vec{a} \) and \( \vec{b} \) are given as 2 and 4, this means \( \|\vec{a}\| = 2 \) and \( \|\vec{b}\| = 4 \). They are essentially the hypotenuses of right-angled triangles built with their respective components.
Dot Product
The dot product is a key concept in vector algebra that produces a scalar value from two vectors. This scalar can help us understand the geometric relationship between the vectors. For two vectors \( \vec{a} = (a_1, a_2) \) and \( \vec{b} = (b_1, b_2) \), the dot product is computed as:
  • \( \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 \)
In the exercise, it's given that \( \vec{a} \cdot \vec{b} = -8 \), indicating a negative dot product. This suggests that the angle between \( \vec{a} \) and \( \vec{b} \) is obtuse (between 90 and 180 degrees), hinting at opposing directions in their vectors. It also provides a crucial piece to solve vector magnitude problems involved in the exercise.
Vector Algebra
Vector algebra involves operations that combine multiple vectors to get another vector. These operations play a crucial role in solving geometry and physics problems and include addition, subtraction, and scaling of vectors.
  • Addition: Adding vectors results in a new vector which combines their individual effects. If \( \vec{a} = (x_1, y_1) \) and \( \vec{b} = (x_2, y_2) \), then \( \vec{a} + \vec{b} = (x_1 + x_2, y_1 + y_2) \).
  • Magnitude of the Sum: The formula to find the magnitude of the sum of two vectors takes into consideration both their individual magnitudes and their dot product: \( \|\vec{a} + \vec{b}\| = \sqrt{\|\vec{a}\|^2 + \|\vec{b}\|^2 + 2\vec{a} \cdot \vec{b}} \).
In this exercise, this formula is used to solve for \( \|\vec{a} + \vec{b}\| \), which beautifully combines these concepts by acknowledging the magnitudes and dot product, resulting in the magnitude of their sum.
Two-dimensional Vectors
Two-dimensional vectors are fundamental in describing motions and forces on a plane. They have an x (horizontal) and a y (vertical) component, making them easy to visualize and useful for many applications in physics and engineering.
  • Represented as \( \vec{v} = (x, y) \), they can point in any direction on a flat surface.
  • They help describe positions, velocities, and forces that have both magnitude and direction.
A two-dimensional vector can be the simplest form of a vector yet powerful for practical problems, just like in this exercise where we work with vectors \( \vec{a} \) and \( \vec{b} \) in the coordinate plane to compute their properties and relationships through vector addition and algebra, employing magnitudes and dot products.

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Most popular questions from this chapter

Find an equation of a plane that satisfies the given conditions. Through (1,3,5) and normal to \(\vec{i}-\vec{j}+\vec{k}\)

Explain what is wrong with the statement. A point \((a, b)\) such that the displacement vector from (1,1) to \((a, b)\) is perpendicular to \(\vec{i}+2 \vec{j}\)

Find an equation of a plane that satisfies the given conditions. Perpendicular to \(5 \vec{i}+\vec{j}-2 \vec{k}\) and passing through (0,1,-1)

Explain what is wrong with the statement. There is only one unit vector perpendicular to two nonparallel vectors in 3 -space.

Compute the angle between the vectors. $$\vec{i} \text { and } 2 \vec{i}+3 \vec{j}-\vec{k}$$
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