Vectors in Three Dimensions Chapter 8 SEC 3 Vectors in Three Dimensions

Vector in 3D Vectors in 3D space are described by ordered triple pairs P(x1, y1, z1). Three number lines that intersect at the zero points To show this on paper we have the x-axis appearing to come out of the paper To locate a point first find x1 on the x-axis, y1 on the y-axis and z1 on the z-axis. Imagine a plane perpendicular to the x-axis at x1 and the same for the other points. The three planes will intersect at point P.

Example 1 Locate the point at (–4, 3,2). x y z (–4, 3, 2)

Ordered Triples Ordered triples can be used to represent vectors. The geometric interpretation is the same as ordered pairs. A directed line from the origin O to P(x, y, z) is called vector corresponding to vector (x, y, z) An extension of the formula for distance give us the distance from the origin to the point (x, y, z) is So the magnitude of vector (x, y, z) is

Example 2 Write the ordered triple that represents the vector from A(–2, –5, 0) to B(3, 1, 8).

Unit Vectors in 3D Three unit vectors are used as components of vectors in space. The unit vectors on the x-, y-, z-axes are respectively, where Vector can be written as

Example 3 Write as the sum of unit vectors for G(–2, –5, 4) and H(1, 5, 6). First express as an ordered triple. Then write the sum of the unit vectors

Perpendicular Vectors Chapter 8 SEC 4 Perpendicular Vectors

Perpendicular Vectors Let be perpendicular vectors, and let be a vector between their terminal points as shown The magnitudes of must satisfy the Pythagorean Theorem.

Inner Products and Dot Products Compare the resulting equation with the original one. if and only if a1b1 + a2b2 = 0. The expression a1b1 + a2b2 = 0 is frequently used in the study of vectors. It is called the inner product of Two vectors are perpendicular if and only if their inner product is zero.

Example 1 Find each inner product if Are any pairs of vectors perpendicular? The inner products of vectors in space is similar to that of vectors in a plane.

Vectors in Space Example 2 Find the inner product of if Are perpendicular? Since the inner product is –1, are not perpendicular.

Chapter 8 Sec 5 Application

Lenny Montana was a former 5-year world heavyweight wrestling champion Lenny Montana was a former 5-year world heavyweight wrestling champion. Suppose Lenny and a tag team partner are each pulling horizontally on the arms of an opponent. Lenny exert a force of 180 pounds due north while his partner exerts a force of 125 due east. a. Draw a labeled diagram that represents the forces. b. Determine the resultant force exerted on the opponent. c. Determine the angle the resultant makes with the east-west axis. Example 1 N W E S 180 lb 180 lb θ 125 lb

Example 2 Justin work for a package delivery service. Suppose that he is pushing a cart full of packages weighing 125 pounds up a ramp 10 feet long at an incline of 20°. Find the work done by gravity as the cart moves the length of the ramp. Assume that friction is not a factor. Let represent the force of gravity or weight. The weight has a force of 125 lbs and it’s direction is down. The unit vector is The application of force is and it’s magnitude is 10 ft. Write use trig to find x and y. y x P(x, y) 20° O Q (0, –125)

Example 2 Justin work for a package delivery service. Suppose that he is pushing a cart full of packages weighing 125 pounds up a ramp 10 feet long at an incline of 20°. Find the work done by gravity as the cart moves the length of the ramp. Assume that friction is not a factor. Apply the formula for determining the work done by gravity. Work done by gravity is negative when a object is lifted. y x P(x, y) 10 20° O Q (0, –125)

Example 3 Ms. Davis is hanging a sigh for her restaurant The sign is supported by two lightweight support bars as shown in the diagram. If the bars make a 30° angles with each other and the sign weighs 200 pounds, what are the magnitudes of the forces exerted by the sign on each support bar? represents force on bar 1, on bar 2, and weight of the sign.

Example 4 A lighting system for a theater is supported equally by two cables suspended from the ceiling. The cables form a 140° angle with each other. If the lighting system weighs 950 pounds, what is the force exerted by each of the cables on the lighting system? 140° x lb x lb 20° 20° 950 lb x lb 475 lb 20° 950 lb

Daily Assignment Chapter 8 Sections 3 – 5 Text Book Pg 503 Pg 509 #13 – 33 Odd; Pg 509 #11 – 19 Odd; Pg 517 #15, 17, 19;