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Chapter 6: Problem 8

Suppose \(\mathbf{u}=(-3,2)\) and \(\mathbf{v}=(-2,-1)\) (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
(a) Draw a Cartesian plane with origin O and label the axes. Represent \(\mathbf{u}\) and \(\mathbf{v}\) as arrows with endpoints A and B respectively. Draw \(\mathbf{v}\) with its tail at A and its endpoint labeled C. The vector OC' represents the sum \(\mathbf{u}+\mathbf{v}\). (b) Using coordinates, compute the sum as \[\mathbf{u}+\mathbf{v} = (-3-2, 2-1) = (-5, 1).\]

Step by step solution

01

Analyze the given vectors

We are given two vectors: \(\mathbf{u}=(-3,2)\) and \(\mathbf{v}=(-2,-1)\). Each vector has an x-component and a y-component, which are the distances it covers along the x-axis and y-axis respectively.
02

Set up a coordinate system

Draw a Cartesian plane (x-axis and y-axis) with the origin, O, at its center. Make sure to label the axes. We will use this plane to represent the vectors graphically.
03

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

Start at the origin and draw an arrow representing the vector \(\mathbf{u}\), with a horizontal component of -3 units to the left and a vertical component of 2 units up. Similarly, start at the origin and draw an arrow representing the vector \(\mathbf{v}\), with a horizontal component of -2 units to the left and a vertical component of -1 units down. Label the endpoints of these arrows as A and B respectively.
04

Draw the vector sum graphically

Place the tail of vector \(\mathbf{v}\) at the tip of vector \(\mathbf{u}\). In other words, draw a vector starting at point A and ending at a point C that has the same length and direction as vector \(\mathbf{v}\). Label this vector as \(\mathbf{v}\). The vector sum \(\mathbf{u}+\mathbf{v}\) is given by the vector starting at the origin O and ending at point C. Draw this vector and label the endpoint as C'.
05

Compute the sum of vectors using coordinates

To compute the sum of the vectors using coordinates, simply add the corresponding x-components and y-components of the two vectors: \[\mathbf{u}+\mathbf{v} = (-3-2, 2-1) = (-5, 1).\]
06

Verify the graphical and coordinate sum

The vector sum of \(\mathbf{u}+\mathbf{v}\) graphically should match the computed sum of the vectors using coordinates, which is \((-5, 1)\). Verify that the vector drawn in Step 4 (OC') has a horizontal component of -5 units to the left and a vertical component of 1 unit up. If the graphical vector and the coordinate sum match, we have correctly computed and illustrated the sum of the vectors.

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