High School: Number and Quantity
Vector and Matrix Quantities HSN-VM.A.3
3. Solve problems involving velocity and other quantities that can be represented by vectors.
Students should understand that verbal problems aren't going to kill them, make them vomit, or cause them any permanent brain damage. Of course, they can always consult their physicians about that, too.
Verbal problems are the real "guts" of most math courses. After all, how often will they have to solve the equation x2 + 8x – 3 = 18? Not too often, right? (With the exception of becoming a math teacher. Well, you know all about that, don't you.)
But verbal problems are a different kettle of fish. They give students the opportunity to put all those equations and logical thought to practical use. Students should be able to take the concepts of vectors and apply them to velocities in real-life scenarios.
Let's assume you own a boat (and hopefully that's a real-life scenario). You're on a river that has a current of 4 mph and your speedometer says you're going 15 mph. If you're going downstream, how fast are you actually going?
Going downstream means that the current is actually pushing you where you want to go, increasing your speed. The magnitude of your boat's vector is 15. The magnitude of the current is 4. The combined magnitude, the sum of those two vectors going in the same exact direction, is 19. That means you're actually going 19 mph.
On the return trip, you're going upstream under the same conditions. Now how fast are you actually going?
Well, you knew it was too good to be true. That same current that helped you make time on the trip downriver is now pushing against the front of your boat. The boat's vector and the current's vector are pointing in opposite directions.
Now your 15 mph magnitude is being added to a -4 magnitude, for a resultant of only 11 mph. Even though your boat's engine is chugging along at 15 mph, you're only achieving 11 mph. The current is slowing you down.
What about more complicated problems where vectors aren't opposite one another? Students should be able to use a right triangle or a coordinate plane to solve slightly more complicated vector word problems, such as finding the velocity of a plane flying due south at 100 mph with a side wind of 20 mph, going due east. (The answer, by the way, is , or approximately 102 mph.)