10.2 Assess Your Understanding

Concepts and Vocabulary

Question

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

Question

Scalars are quantities that have only ______.

Question

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

Question

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

Skill Building

Question

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

In Problems 6–14, use the vectors in the above figure to graph each of the following vectors:

Question

\(2\mathbf{v}\)

Question

\(-2\mathbf{v}\)

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

In Problems 15–22, use the vectors in the figure below.

Question

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

Question

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

703

Question

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 \(\mathbf{K}\).

Question

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 \(\mathbf{D}\).

Question

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

Question

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

Applications and Extensions

Question

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 \(\mathbf{F}_{1}+\mathbf{F}_{2}.\)

Question

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 \(\mathbf{F}_{1}+\mathbf{F}_{2}.\)

Question

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.

Question

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.

Question

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 \( \mathbf{w}\).

Question

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 \(\mathbf{w}\).