In the following figure, the magnitudes of the vectors are \(|\vec{a}| = 8\) and \(|\vec{b}| = 4.5\text{.}\) Assume that \(\vec{c} = \vec{a} + \vec{b}\) and \(\vec{d} = \vec{a} - \vec{b}\text{.}\)
Figure1.11.1.Two vectors.
Determine the magnitude of the vectors \(\vec{c}\) and \(\vec{d}\text{?}\) What is the angle to each vector from the positive \(x\)-axis?
Answer.
\(\vec{c}\text{:}\)\(10.2\) at \(120^o\)
\(\vec{d}\text{:}\)\(8.2\) at \(167^o\)
Calculation1.11.2.Two Vectors II.
In the following figure, the magnitudes of the vectors are \(|\vec{a}| = 5\) and \(|\vec{b}| = 5\text{.}\) Assume that \(\vec{c} = \vec{a} + \vec{b}\) and \(\vec{d} = \vec{a} - \vec{b}\text{.}\)
Figure1.11.2.Two vectors.
Determine the magnitude of the vectors \(\vec{c}\) and \(\vec{d}\text{?}\) What is the angle to each vector from the positive \(x\)-axis?
Answer.
\(\vec{c}\text{:}\)\(4\) at \(-42.5^o\)
\(\vec{d}\text{:}\)\(8.87\) at \(227.5^o\)
Calculation1.11.3.Vector Contest.
Three vectors add together to equal \(0\text{.}\) One vector has magnitude \(3\) and points in the positive \(x\)-direction; a second vector has magnitude \(5\) and points at \(120^o\) from the positive \(x\)-axis. Determine the third vector as a magnitude and direction.
Answer.
\(4.4\) at \(-97^o\)
Calculation1.11.4.Sums and Differences of Vectors.
Use the figure to determine a single vector that is equivalent to the given summation of vectors in each of the cases below. (The notation \(\vec{PQ}\) represents a vector pointing from point \(P\) to point \(Q\text{.}\))
Figure1.11.3.Four points.
\(\displaystyle \vec{PQ} + \vec{QR}\)
\(\displaystyle \vec{RP} + \vec{PS}\)
\(\displaystyle \vec{QS} + \vec{PS}\)
\(\displaystyle \vec{RS} + \vec{SP} + \vec{PQ}\)
Answer.
\(\displaystyle \vec{PR}\)
\(\displaystyle \vec{RS}\)
\(\displaystyle \vec{QP}\)
\(\displaystyle \vec{RQ}\)
SubsectionPosition and Displacement Vectors
Calculation1.11.5.Punting a Football.
A kicker punts a football from the very center of the field to the sideline \(42\) yards downfield. What is the magnitude of the net displacement of the ball in yards?
Tip.
A football field is \(100\) yards long and \(55\) yards wide.
Answer.
\(50.2\) yards
Calculation1.11.6.The Flying Saucer.
A certain flying saucer is initially located at a position \(\vec{r}_i = 400 \hat{x} + 350 \hat{y} \mathrm{~m}\text{.}\) After a few minutes it has moved to a location \(\vec{r}_f = 650 \hat{x} - 800 \hat{y} \mathrm{~m}\text{.}\) What is the displacement of the flying saucer?
A park has a circular pond with a radius of \(100 \mathrm{~m}\text{.}\) Benny starts at its westernmost point, then waddles counterclockwise around the pond until he is at its northernmost point. What is the magnitude and direction of Benny’s change in position?
Answer.
\(141 \mathrm{~m}\) northeast
ReferencesReferences
[1]
Practice activities provided by BoxSand: https://boxsand.physics.oregonstate.edu/welcome.