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True or False A vector is an object that has magnitude and direction.

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Scalars are quantities that have only ______.

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The product of a scalar \(a\) and a vector \(\mathbf{v}\) is called a(n) _____ ______ of \(\mathbf{v}\).

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Multiple Choice The vectors \(-\mathbf{v}\) and \(\mathbf{v}\) have [(a) the same, (b) different] magnitude and the [(c) same, (d) opposite] direction.

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State which of the following are scalars and which are vectors:

1. Volume
2. Speed
3. Force
4. Work
5. Mass
6. Distance
7. Age
8. Velocity

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\(2\mathbf{v}\)

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\(-2\mathbf{v}\)

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\(\mathbf{v} + \mathbf{w}\)

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\(\mathbf{v}-\mathbf{w}\)

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\(\mathbf{w}-\mathbf{v}\)

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\(\mathbf{v}-2\mathbf{w}\)

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\((\mathbf{v+w}) +3\mathbf{u}\)

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\(\mathbf{v}+( \mathbf{w}+3\mathbf{u}) \)

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\(2\,\mathbf{u}-\dfrac{1}{3}(\mathbf{v}-\mathbf{w})\)

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Find the vector \(\mathbf{x}\) if \(\mathbf{x}+\mathbf{B}=\mathbf{F}\).

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Find the vector \(\mathbf{x}\) if \(\mathbf{x}+\mathbf{K}= \mathbf{C}\).

703

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Write \(\mathbf{C}\) in terms of \(\mathbf{E}\) , \( \mathbf{D}\) , and \(\mathbf{F}\).

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Write \(\mathbf{G}\) in terms of \(\mathbf{C}\) , \( \mathbf{D}\) , \(\mathbf{E}\), and \(\mathbf{K}\).

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Write \(\mathbf{E}\) in terms of \(\mathbf{G}\) , \( \mathbf{H}\) , and \(\mathbf{D}\).

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Write \(\mathbf{E}\) in terms of \(\mathbf{A}\) , \( \mathbf{B}\) , \(\mathbf{C}\) , and \(\mathbf{D}\).

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What is \(\mathbf{A}+\mathbf{B}+\mathbf{K}+\mathbf{G}\)?

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What is \(\mathbf{A}+\mathbf{B}+\mathbf{C}+\mathbf{H}+\mathbf{G}\)?

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Resultant Force Two forces \(\mathbf{F}_{1}\) and \( \mathbf{F}_{2}\) act on an object shown in the figure. Graph the vector representing the resultant force; that is, find \(\mathbf{F}_{1}+\mathbf{F}_{2}.\)

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Resultant Force Two forces \(\mathbf{F}_{1}\) and \(\mathbf{F} _{2}\) act on an object shown in the figure. Graph the vector representing the resultant force; that is, find \(\mathbf{F}_{1}+\mathbf{F}_{2}.\)

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Air Travel An airplane is flying due north at a constant airspeed of 560 \(\rm{mi/h}\). There is a wind \(75 \,\rm{mi/h}\) blowing from the east.

1. Draw vectors representing the velocities of the airplane and the wind.
2. Draw the vector representing the airplane’s ground speed.
3. Interpret the result.

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Air Travel An airplane maintains a constant airspeed of \(500 \,\rm{km/h}\) headed due west. There is a tail wind blowing at \(500 \,\rm{km/h}\).

1. Draw vectors representing the velocities of the airplane and the tail wind.
2. Draw the vector representing the airplane’s ground speed.
3. Interpret the result.

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Suppose \(\mathbf{v}\) and \(\mathbf{w}\) are nonzero vectors represented by arrows with the same initial point, and that the terminal points of \(\mathbf{v}\) and \(\mathbf{w}\) are \(P\) and \(Q\), respectively. Suppose the vector \(\mathbf{u}\) is represented by an arrow from the initial point of \(\mathbf{v}\) to the midpoint of the directed line segment \( \overrightarrow{\it PQ}\). Write \(\mathbf{u}\) in terms of \(\mathbf{v}\) and \( \mathbf{w}\).

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Find nonzero scalars \(a\) and \(b\) so that \[ \begin{equation*} 4\mathbf{v}+a(\mathbf{v}-\mathbf{w})+b(\mathbf{v}+\mathbf{w})=\mathbf{0} \end{equation*} \]

for every pair of vectors \(\mathbf{v}\) and \(\mathbf{w}\).