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Chapter 8: Problem 25

Given vectors u and v, find: (a) 2u (b) 2u+3v (c) \(\mathbf{v - 3 u .}\) $$\mathbf{u}=2 \mathbf{i}, \mathbf{v}=\mathbf{i}+\mathbf{j}$$

Short Answer

Expert verified
2u = 4i, 2u + 3v = 7i + 3j, v - 3u = -5i + j.

Step by step solution

01

Write the given vectors

Given: \(\textbf{u} = 2\textbf{i}\) \(\textbf{v} = \textbf{i} + \textbf{j}\)
02

Find 2u

To find \(2\textbf{u}\), multiply each component of \(\textbf{u}\) by 2: \(2 \textbf{u} = 2 \times 2\textbf{i} = 4\textbf{i}\)
03

Find 2u + 3v

Multiply \(\textbf{v}\) by 3 and add to \(2\textbf{u}\): \(3\textbf{v} = 3(\textbf{i} + \textbf{j}) = 3\textbf{i} + 3\textbf{j}\)So, \(2\textbf{u} + 3\textbf{v} = 4\textbf{i} + 3\textbf{i} + 3\textbf{j}\)Combine like terms: \(2\textbf{u} + 3\textbf{v} = 7\textbf{i} + 3\textbf{j}\)
04

Find v - 3u

Multiply \(\textbf{u}\) by 3 and subtract from \(\textbf{v}\): \(3\textbf{u} = 3 \times 2\textbf{i} = 6\textbf{i}\)So, \(\textbf{v} - 3\textbf{u} = (\textbf{i} + \textbf{j}) - 6\textbf{i}\)Combine like terms: \(\textbf{v} - 3\textbf{u} = \textbf{i} - 6\textbf{i} + \textbf{j} = -5\textbf{i} + \textbf{j}\)

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Key Concepts

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

vector addition
Vector addition is the process of combining two or more vectors to form a new vector. When adding vectors, each corresponding component is added separately. In essence, you simply add the x-components together and the y-components together.
Let's look at an example involving vectors \(\textbf{u} = 2\textbf{i}\) and \(\textbf{v} = \textbf{i} + \textbf{j}\). To find \(\textbf{2u} + 3\textbf{v}\), follow these steps:
  • Multiply \(\textbf{u}\) by 2: \(\textbf{2u} = 4\textbf{i}\)
  • Multiply \(\textbf{v}\) by 3: \(\textbf{3v} = 3(\textbf{i} + \textbf{j}) = 3\textbf{i} + 3\textbf{j}\)
  • Add the resulting vectors: \(\textbf{2u} + \textbf{3v} = 4\textbf{i} + 3\textbf{i} + 3\textbf{j} = 7\textbf{i} + 3\textbf{j}\)
The final result \(\textbf{7i + 3j}\) is the sum of two vectors. Remember to always align the corresponding components before adding them.
scalar multiplication
Scalar multiplication involves scaling a vector by a scalar (a real number). Each component of the vector is multiplied by this scalar. This process either stretches or shrinks the vector, but does not change its direction unless the scalar is negative, in which case the direction is reversed.
Consider the vector \(\textbf{u} = 2\textbf{i}\). If we want to find \(2\textbf{u}\), we multiply each component of \(\textbf{u}\) by 2:
  • Multiplying the \(\textbf{i}\)-component: \(\textbf{2} \times 2\textbf{i} = 4\textbf{i}\)
The result \(\textbf{4i}\) is a vector pointing in the same direction as \(\textbf{u}\), but twice as long. This concept is fundamental when manipulating vectors in mathematics and physics.
vector subtraction
Vector subtraction is the process of finding the difference between two vectors. It can be thought of as adding the first vector to the negative of the second vector. To subtract vectors, you subtract each corresponding component separately.
Let's look at how to find \(\textbf{v} - 3\textbf{u}\) with \(\textbf{u} = 2\textbf{i}\) and \(\textbf{v} = \textbf{i} + \textbf{j}\):
  • Multiply \(\textbf{u}\) by 3: \(\textbf{3u} = 3 \times 2\textbf{i} = 6\textbf{i}\)
  • Subtract \(\textbf{3u}\) from \(\textbf{v}\): \(\textbf{v} - \textbf{3u} = (\textbf{i} + \textbf{j}) - 6\textbf{i} = \textbf{i} - 6\textbf{i} + \textbf{j} = -5\textbf{i} + \textbf{j}\)
The result \(\textbf{-5i + j}\) shows how subtraction affects both direction and magnitude of the vectors involved. Practicing these operations will make working with vectors much more intuitive.

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Most popular questions from this chapter

Use the law of sines to prove that each statement is true for any triangle \(A B C\), with corresponding sides \(a, b,\) and \(c\) $$\frac{a+b}{b}=\frac{\sin A+\sin B}{\sin B}$$

For each equation, find an equivalent equation in rectangular coordinates, and graph. $$r=\frac{2}{1-\cos \theta}$$

Solve each problem. The polar equation $$r=\frac{a\left(1-e^{2}\right)}{1+e \cos \theta}$$ can be used to graph the orbits of the satellites of our sun, where \(a\) is the average distance in astronomical units from the sun and \(e\) is a constant called the eccentricity. The sun will be located at the pole. The table lists the values of \(a\) and \(e\). (a) Graph the orbits of the four closest satellites on the same polar grid. Choose a viewing window that results in a graph with nearly circular orbits. (b) Plot the orbits of Earth, Jupiter, Uranus, and Pluto on the same polar grid. How does Earth's distance from the sun compare to the others' distances from the sun? (c) Use graphing to determine whether or not Pluto is always farthest from the sun. $$\begin{array}{|c|c|c|}\hline \text { Satellite } & a & e \\\\\hline \text { Mercury } & 0.39 & 0.206 \\\\\hline \text { Venus } & 0.78 & 0.007 \\\\\hline \text { Earth } & 1.00 & 0.017 \\\\\hline \text { Mars } & 1.52 & 0.093 \\\\\hline \text { Jupiter } & 5.20 & 0.048 \\\\\hline \text { Saturn } & 9.54 & 0.056 \\\\\hline \text { Uranus } & 19.20 & 0.047 \\\\\hline \text { Neptune } & 30.10 & 0.009 \\\\\hline \text { Pluto } & 39.40 & 0.249 \\\\\hline\end{array}$$

Give a complete graph of each polar equation. Also identify the type of polar graph. $$r^{2}=4 \sin 2 \theta$$

For each equation, find an equivalent equation in rectangular coordinates, and graph. $$r=\frac{3}{1-\sin \theta}$$
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