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Chapter 6: Problem 7

Suppose \(\mathbf{u}=(2,1)\) and \(\mathbf{v}=(3,1)\) (a) Draw a figure illustrating the sum of \(\mathbf{u}\) and \(\mathbf{v}\) as arrows. (b) Compute the sum \(\mathbf{u}+\mathbf{v}\) using coordinates.

Short Answer

Expert verified
The sum of the vectors \(\mathbf{u}=(2,1)\) and \(\mathbf{v}=(3,1)\) is \(\mathbf{u}+\mathbf{v}=(5,2)\).

Step by step solution

01

Draw the vectors \(\mathbf{u}\) and \(\mathbf{v}\)

To draw the vectors, use a coordinate plane. Draw an arrow from the origin (0,0) to the point represented by the coordinates of each vector. For \(\mathbf{u}\), draw an arrow from (0,0) to (2,1) and label it \(\mathbf{u}\). For \(\mathbf{v}\), draw an arrow from (0,0) to (3,1) and label it \(\mathbf{v}\). Next, let's draw the sum of the vectors as an arrow:
02

Draw the vector \(\mathbf{u}+\mathbf{v}\)

To draw the sum of \(\mathbf{u}\) and \(\mathbf{v}\), place the initial point of \(\mathbf{v}\) at the terminal point of \(\mathbf{u}\) (i.e., place the tail of \(\mathbf{v}\) at the head of \(\mathbf{u}\). Then draw an arrow from the origin (0,0) to the terminal point of the translated \(\mathbf{v}\) and label it \(\mathbf{u}+\mathbf{v}\). Now let's compute the sum of the vectors:
03

Calculate the sum of the vectors \(\mathbf{u}+\mathbf{v}\)

To compute the sum of the vectors \(\mathbf{u}\) and \(\mathbf{v}\), we simply add their respective coordinates: \(\mathbf{u}+\mathbf{v} = (u_x + v_x, u_y + v_y) = (2+3, 1+1) = (5,2)\) The sum of the vectors \(\mathbf{u}\) and \(\mathbf{v}\) is \(\mathbf{u}+\mathbf{v}=(5,2)\).

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Most popular questions from this chapter

Write each expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$ (5+\sqrt{6} i)^{2} $$

Write each expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$ (2+3 i)^{3} $$

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