Cholkar WYLA INT 4 ___/___/___ Name______
U2L1INV1 / How can vectors be represented geometrically with directed line segments?How can vectors and scalar multiples of vectors be used to model navigation routes?
HW # 10 / CYU pg. 108; pg. 117 # 1; pg. 126 # 24, 25 [CYU, 24]
Do Now / SAT QUESTION OF THE DAY!
{globe, protractors, nautical mile rulers, compasses, CPMP Tools}
INVESTIGATION: NAVIGATION: WHAT DIRECTION AND HOW FAR? (pg. 103)
My role for this investigation ______
THINK ABOUT THIS SITUATION:
Suppose you wanted to map out a route that involved sailing 3 km west from Bayview Harbor to Presque Island, then 6 km south to Rudy Point, and then 5km southeast to Pleasant Bay.
a. How could you represent the planned route geometrically? ______
______
b. How could you represent a direct sailing route from Bayview Harbor to Pleasant Bay?
______
c. How could you estimate the length of the route in Part b? ______
How would you describe its direction? ______
d. How would a northeast water current affect the path along which you would steer the boat to maintain the route in Part b?
______
1. As a class, let’s examine this copy of the nautical chart.
a. Use the nautical mile (nm) scale to the right of the chart to find the distance from the “SH” buoy to the “GP” buoy. Measure between the centers of the circles that mark the buoys.
______
b. What do you think the scales along the top and right side of the chart represent?
______
c. What other scale on this chart can be used to measure nautical miles? ______
______
What does a nautical mile represent based on this scale? ______
d. A nautical mile is 6,076,1033 feet. How does a nautical mile compare to a statute mile(regular mile)?
______
2. a. On the diagram below, mark and label point P to represent a boat that is 3 nautical miles from the “3” bell and is headed at an angle of 290°. What buoy is nearest to P?
______
b. Draw an arrow from the “SH” buoy to the “6” buoy.
What is the direction in degrees?
______
What is the distance in nautical miles?
______
c. What are the direction and distance of the path from the
“6” buoy to the center of the mouth of the channel at Stone
Harbor?
______
Why are arrows particularly useful representations for
nautical paths?
______
3. Vectors: ______
A vector with a length of 1” and direction of 45° is shown at the right.
a. Accurately draw arrows representing vectors with the following characteristics.
i.
ii. iii.
b. Draw an arrow for each vector described. State what length you chose to represent 1 knot and what length you chose to represent 1 mph.
i.
ii.
iii.
c. Send your spy to another group to compare the arrows you drew in Parts a and b.
Vectors can be denoted in several ways: or
4. Equal Vectors: ______
Explain why the following method provides a geometric test for the equality of the vectors .
______
______
5. A fishing boat leaves the mouth of the Stone Harbor channel trolling on a heading due north at a speed of 1.5 knots (nautical miles per hour).
a. On the copy of the nautical chart above, sketch the vector representing the distance and direction traveled from the middle of the channel opening during the first hour.
b. Use the vector in Part a to determine the vector for a 2-hour trip at the same speed and in the same direction. Sketch this vector. Label it 2.
c. Sketch and label a vector that locates the fishing boat at the end of 20 minutes. At the end of 2.5 hours.
d. Now sketch another vector that has the same length at 2 but is not equal to 2, and another vector that is equal to 2. Compare your vectors with those of another group.
e. In general, how would you sketch a vector that was a positive number k times a given vector?
______
f. How are the lengths and directions of these two vectors related?
______
6. Suppose another boat begins a trip at the same point at the mouth of the channel at Stone Harbor headed at a direction of 20° and at a speed of 2 knots.
a. Sketch the vector showing the approximate position at the end of the first hour.
b. Suppose the boat returns to the harbor along the same route at the same speed. Sketch the return vector and give its magnitude and direction.
c. The word “opposites” can be used to denote the vectors in Parts a and b. How is the word “opposite” descriptive of the relationship between the two vectors?
______
d. Sketch a vector opposite to vector in Part a with initial point at the “3” bell. Give its magnitude and direction.
7. Scalar Multiple: ______
(ie: , is the scalar multiple of )
For vector shown to the right, the opposite of vector ,
denoted is shown at the right. The scalar multiple when
k < 0 is shown at the far right.
a.
______
b.
c.
______
Lesson Summary / In this investigation, you explored how vectors-quantities with magnitude and direction-can be represented geometrically by arrows.Math Toolkit Vocabulary: vectors, scalars, equal vectors, scalar multiple
Cholkar WYLA INT 4 ___/___/___ Name______
HW # 10
CYU pg. 108