Vectors 9.7 Chapter 9 Right Triangles and Trigonometry Section 9.7 Vectors Find the magnitude and the direction of a vector. Add vectors.

Vectors 9.7 Lesson 6 Contents Key Concepts Example 1Write Vectors in Component Form Example 2Magnitude and Direction of a Vector Example 3Add Vectors Example 4Solve Problems Using Vectors Homework Find the magnitude and the direction of a vector. Add vectors.

Vectors 9.7 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 units to the right 3 units up 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.

Vectors 9.7 Length of Vector Angle measured from horizontal

Vectors 9.7

Vectors 9.7 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 = A B The direction is the angle measured from the horizontal to the vector

Vectors 9.7 Example 6-1a Write the component form of Draw it in standard position Find the magnitude and the direction of a vector.

Vectors 9.7 Example 6-1a 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.

Vectors 9.7 Example 6-1a 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.

Vectors 9.7 Example 6-1b Write the component form of Draw it in standard position Answer: Find the magnitude and the direction of a vector.

Vectors 9.7 Example 6-2a 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.

Vectors 9.7 Example 6-2a 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.

Vectors 9.7 Example 6-2a Simplify. Substitution Use a calculator. tan S Find the magnitude and the direction of a vector. ???????????????????? ???????????????????? ???????????

Vectors 9.7 Example 6-2a 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.

Vectors 9.7 Example 6-2b 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.

Vectors 9.7 Example 6-4a Add a and b. Add the vectors a = b = Add vectors. Sum of Two Vectors: The sum of u = and v = is u + v =.

Vectors 9.7 Example 6-4b Find the Sum of Answer: Add vectors.

Vectors 9.7 Example 6-5a 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.

Vectors 9.7 Example 6-5a 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.

Vectors 9.7 Example 6-5a 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.

Example 6-5a Find the magnitude and the direction of a vector. The magnitude of the vector represents the planes speed. Use the Distance Formula. Reduce

Vectors 9.7 Example 6-5a Find the magnitude and the direction of a vector.

Vectors 9.7 Example 6-5a Answer:If The Resultant direction of the plane is about North 4.6° East Traveling at mph Answer:If The Resultant direction of the plane is about East 85.4° North Traveling at mph Find the magnitude and the direction of a vector.

Vectors 9.7 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? Example 6-5b 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.