vectors 9.7 chapter 9 right triangles and trigonometry section 9.7 vectors find the magnitude and...
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Vectors9.7
Chapter 9Right Triangles and Trigonometry
Section 9.7
Vectors
Find the magnitude and the direction of a vector.
Add vectors.
Vectors9.7
Key Concepts
Example 1 Write Vectors in Component Form
Example 2 Magnitude and Direction of a Vector
Example 3 Add Vectors
Example 4 Solve Problems Using Vectors
Homework
Find the magnitude and the direction of a vector.
Add vectors.
Vectors9.7
Another way to describe a translation is by using a vector. A vector is a quantity that has both direction and magnitude, or size, and is represented by an arrow drawn between two points.
P
Q
The diagram shows a vector.
The initial point, or starting point, of the vector is P.
The terminal point, or ending point, is Q.
The vector is named PQ, which is read as “ vector PQ.”
The horizontal component of PQ is 5 and the vertical component is 3.
5 unitsto the right
3 unitsup
The component form of a vector combines the horizontal and vertical components. So, the component form of PQ is 5, 3 .
Find the magnitude and the direction of a vector.
Vectors9.7
Length of Vector
Angle measured
from horizontal
Vectors9.7
Vectors9.7
Vector in standard Position
A
B
To write AB in standard position move A to be on the origin and redraw the vector with the same horizontal and vertical components.
AB = <5, 4>
A
B
The direction is the angle measured from the horizontal to the vector
Vectors9.7
Write the component form of Draw it in standard position
Find the magnitude and the direction of a vector.
Vectors9.7
Find the change of x values and the corresponding change in y values.
Component form of vector
Simplify.
Find the magnitude and the direction of a vector.
Vectors9.7
Find the magnitude and the direction of a vector.
Answer: Because the magnitude and direction of a vector are not changed by translation, the vector can be written in standard position.
Vectors9.7
Write the component form of Draw it in standard position
Answer:
Find the magnitude and the direction of a vector.
Vectors9.7
Find the magnitude and direction of for S(–3, –2) and T(4, –7).
Find the magnitude.
Distance Formula
Simplify.
Use a calculator.
Find the magnitude and the direction of a vector.
Vectors9.7
Graph to determine how to find the direction. Draw a right triangle that has as its hypotenuse and an acute angle at S.
Find the magnitude and the direction of a vector.
Vectors9.7
Simplify.
Substitution
Use a calculator.
tan S
Find the magnitude and the direction of a vector.
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Vectors9.7
A vector in standard position that is equal to forms a –35.5° degree angle with the positive x-axis in the fourth quadrant. So it forms a angle with the positive x-axis.
Answer: has a magnitude of about 8.6 units and a direction of about East 35.5° South.
Find the magnitude and the direction of a vector.
Answer: has a magnitude of about 8.6 units and a direction of about East 324.5° North.
Vectors9.7
Find the magnitude and direction of for A(2, 5) and B(–2, 1).
Answer: 5.7; 225°
Find the magnitude and the direction of a vector.
Vectors9.7
Add a and b.
Add the vectors a = <-4, 2> b = <2, 3>
Add vectors.
Sum of Two Vectors: The sum of u = < a1, b1 > and v = < a2, b2> is u + v = <a1 + a2, b1 + b2>.
Vectors9.7
Find the Sum of
Answer:
Add vectors.
Vectors9.7
CANOEING Suppose a person is canoeing due east across a river at 4 miles per hour. If the river is flowing south at 3 miles an hour, what is the resultant direction and velocity of the canoe?
The initial path of the canoe is due east, so a vector representing the path lies on the positive x-axis 4 units long. The river is flowing south, so a vector representing the river will be parallel to the negative y-axis 3 units long. The resultant path can be represented by a vector from the initial point of the vector representing the canoe to the terminal point of the vector representing the river.
Find the magnitude and the direction of a vector.
Vectors9.7
Use the Pythagorean Theorem.Pythagorean Theorem
Simplify. Take the square root of each side.
The resultant velocity of the canoe is 5 miles per hour.Use the tangent ratio to find the direction of the canoe.
Use a calculator.
Find the magnitude and the direction of a vector.
Vectors9.7
The resultant direction of the canoe is about 36.9° south of due east.
Answer: Therefore, the resultant vector is 5 miles per hour at 36.9° south of due east.
Find the magnitude and the direction of a vector.
Find the magnitude and the direction of a vector.
The magnitude of the vector represents the planes speed.
Use the Distance Formula.
62910c Reduce
Vectors9.7
Find the magnitude and the direction of a vector.
Vectors9.7
Answer: If The Resultant direction of the plane is about
North 4.6° East Traveling at 250.8 mph
Answer: If The Resultant direction of the plane is about
East 85.4° North Traveling at 250.8 mph
Find the magnitude and the direction of a vector.
Vectors9.7
KAYAKING Suppose a person is kayaking due east across a lake at 7 miles per hour.
a. If the lake is flowing south at 4 miles an hour, what is the resultant direction and velocity of the canoe?the resultant direction and velocity of the kayak?
Answer: Resultant direction is about 29.7° south of due east; resultant velocity is about 8.1 miles per hour.
Find the magnitude and the direction of a vector.