Exercises
- If you start at the initial point \((7,1)\) and travel to the terminal point \((2,5),\) along what vector did you travel?
- Let \(A\) be the point \((2,-9)\) and let \(B\) be the point \((3,4)\) What are the components of the vector \(\overrightarrow{AB}?\) What are the components of the vector \(\overrightarrow{BA}?\)
- If you start at the initial point \((-2,7)\) and travel along a vector \(\langle -1, 3 \rangle,\) what are the coordinates of the terminal point?
- Given points \(A\) and \(B\) such that the coordinates of \(A\) are \((-3,11)\) and such that \(\overrightarrow{AB} = \bigl\langle 2,-6 \bigr\rangle\) what must the coordinates of \(B\) be?
- Let \(\bm{v} = \langle 3,2 \rangle.\) What is the magnitude of \(\bm{v}?\) What is \(\bm{\hat{v}},\) the unit vector in the direction of \(\bm{v}?\)
-
Sketch the vectors \(\bm{u} = \langle 1,3 \rangle\)
and \(\bm{v} = \langle -4,2 \rangle\) in the \(xy\)-plane.
Then sketch these vectors,
and calculating them explicitly in terms of their components.
\( \bm{u} + \bm{v} \)\( -\bm{u} \)\( \bm{u} - \bm{v} \)\( 4\bm{u} \)\( 2\bm{u} - 3\bm{v} \)\( \bm{v} + \langle 7,-2 \rangle \)\( \bm{u} + 3\mathbf{j} \)
- What angle does \(\bm{u} = \langle 1,3 \rangle\) make with the positive \(x\)-axis?
- What angle does \(\bm{v} = \langle -4,2 \rangle\) make with the positive \(y\)-axis?
- What is the vector of magnitude \(15\) that frames an angle of \(63°\) with the positive \(x\)-axis?
- What is the vector of magnitude \(8\) that frames an angle of \(148°\) with the positive \(x\)-axis?
- If \(|\bm{u}| = 5\) and \(|\bm{v}| = 7\) what, if anything, can you conclude about \(|\bm{u} + \bm{v}|?\)
Problems & Challenges
- Suppose an airplane plans to travel 456 mph with a heading of N30°W. How do you describe this velocity/trajectory as a vector? I.e. what are the east/west and north/south components of the plane’s velocity?
- Suppose an airplane has a velocity vector of \(\langle 234, 389 \rangle\) mph. What is the plane’s heading as an angle? What is the plane’s speed?
- Suppose an airplane has a velocity vector of \(\langle 234, 389 \rangle\) mph, but there is a steady 40 mph wind blowing from the southeast. What is the course of the plane as a vector? What is the actual speed of the plane?
- Suppose an airplane is flying at a constant speed of 430 mph and has an intended course due east-northeast. However there is a 40 mph wind blowing from the north. What should the plane’s heading be to stay on course?
- Suppose an airplane has a velocity vector of \(\langle 234, 389 \rangle\) mph, but due to the effects of a heavy wind, actually has a course of \(\langle 255, 377 \rangle\) mph. What is the velocity vector of the wind? What is the speed of the wind?
- Suppose you must swim across a straight river that is 20 ft wide, and you want to end up at the spot directly across the river from where you are now. The river has a constant current speed of 3 ft/s, but luckily you can swim at about 5 ft/s. In what direction do you need to swim to counteract the current to ensure you land directly across the river?
- Devise a formula for the distance from a point \(P\) to a line \({Ax\!+\!By\!=\!C.}\)