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Over Lesson 8-1 5–Minute Check 1 Determine the magnitude and direction of the resultant of the vector sum described as 25 miles east and then 47 miles south. A.53.2 miles, N28°E B.53.2 miles, S62°E C.53.2 miles, S28°E D.72 miles, S28°E 2 of 37

Over Lesson 8-1 5–Minute Check 3 Which of the following represents a vector quantity? A.a car driving at 55 miles per hour B.a cart pulled up a 30° incline with a force of 40 newtons C.the temperature of a cup of coffee D.wind blowing at 30 knots 3 of 37

A vector whose initial point is at the origin (0, 0) can be uniquely represented by the coordinates of its terminal point (v 1, v 2 ). This is the component form of a vector v, written as 4 of 37

Key Concept 1 It is terminal minus initial 5 of 37

Example 1 Express a Vector in Component Form Find the component form of with initial point A(1, –3) and terminal point B(1, 3). =Component form = = 6 of 37

Example 1 Find the component form of given initial point A(–4, –3) and terminal point B(5, 3). 7 of 37

Key Concept 2 8 of 37

Example 2 Find the Magnitude of a Vector Find the magnitude of with initial point A(1, –3) and terminal point B(1, 3). 9 of 37

Example 2 Find the magnitude of given initial point A(4, –2) and terminal point B(–3, –2). 10 of 37

Key Concept 3 11 of 37

Example 3 Operations with Vectors A. Find 2w + y for w =, y =, and z =. 2w + y = 2 + = + = or 12 of 37

Example 3 Operations with Vectors B. Find 3y – 2z for w =, y =, and z =. 3y – 2z= 3y + (– 2z) = 3 + (–2) = + = or 13 of 37

Example 3 Find 3v + 2w for v = and w = 14 of 37

In many applications of vectors, it is useful to find a unit vector that has the same direction as a given nonzero vector v. To do this, you can divide v by its length to obtain Unit vector in direction of v 15 of 37

Example 4 Find a Unit Vector with the Same Direction as a Given Vector Find a unit vector u with the same direction as v =. or 16 of 37

Example 4 Find a Unit Vector with the Same Direction as a Given Vector Therefore, u =. 17 of 37

Example 4 Find a Unit Vector with the Same Direction as a Given Vector Check Since u is a scalar multiple of v, it has the same direction as v. Verify that the magnitude of u is of 37

Example 4 Find a unit vector u with the same direction as w =. 19 of 37

The unit vectors and are called the standard unit vectors and are denoted by and 20 of 37

Unit Vectors These vectors can be used to represent any vector as follows. The scalars v 1 and v 2 are called the horizontal and vertical components of v, respectively. The vector sum is called a linear combination of the vectors i and j. Any vector in the plane can be written as a linear combination of the standard unit vectors i and j. 21 of 37

Example 5 Write a Unit Vector as a Linear Combination of Unit Vectors First, find the component form of. Let be the vector with initial point D(–3, –3) and terminal point E(2, 6). Write as a linear combination of the vectors i and j. = 22 of 37

Example 5 Answer:5i + 9j Write a Unit Vector as a Linear Combination of Unit Vectors Then, rewrite the vector as a linear combination of the standard unit vectors. = Component form = 5i + 9j = ai + bj 23 of 37

Example 5 Let be the vector with initial point D(–4, 3) and terminal point E(–1, 5). Write as a linear combination of the vectors i and j. 24 of 37

HW: PG odds, odds, 21, 23, odds

Example 6 Find the component form of the vector v with magnitude 7 and direction angle 60°. Find Component Form 26 of 37

Example 6 Find Component Form Check Graph v = ≈. The measure of the angle v makes with the positive x-axis is about 60° as shown, and |v| =. 27 of 37

Example 6 Find the component form of the vector v with magnitude 12 and direction angle 300°. A. B. C. D. 28 of 37

Example 7 Direction Angles of Vectors A. Find the direction angle of p = to the nearest tenth of a degree. 29 of 37

Example 7 Direction Angles of Vectors Answer: 77.5° So the direction angle of vector p is about 77.5°, as shown below. 30 of 37

Example 7 Direction Angles of Vectors B. Find the direction angle of r = –7i + 2j to the nearest tenth of a degree. 31 of 37

Example 7 Direction Angles of Vectors Answer: 164.1° Since r lies in Quadrant II as shown below, θ = 180 – 15.9°or 164.1°. 32 of 37

Example 7 Find the direction angle of p = to the nearest tenth of a degree. 33 of 37

Example 8 SOCCER A soccer player running forward at 7 meters per second kicks a soccer ball with a velocity of 30 meters per second at an angle of 10° with the horizontal. What is the resultant speed and direction of the kick? Since the soccer player moves straight forward, the component form of his velocity v 1 is. Use the magnitude and direction of the soccer ball’s velocity v 2 to write this vector in component form. Applied Vector Operations 34 of 37

Example 8 v 2 = Component form of v 2 = |v 2 | = 30 and θ = 10° ≈ Simplify. Add the algebraic vectors representing v 1 and v 2 to find the resultant velocity, vector r. r= v 1 + v 2 Resultant vector = + Substitution = Vector Addition Applied Vector Operations 35 of 37

Example 8 Applied Vector Operations The magnitude of the resultant is |r| = or about Next find the resultant direction θ. 36 of 37

Example 8 Answer:36.9 m/s; 8.1° Applied Vector Operations Therefore, the resultant velocity of the kick is about 36.9 meters per second at an angle of about 8.1° with the horizontal. 37 of 37

HW: PG , odds, odds, of 37