Vectors. Examples of vectors include: displacement, velocity, acceleration, and force. Examples of scalars include: distance, speed, time, and volume.

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1 Math 150 Prof. Beydler 7.4/7.5 Notes Page 1 of 6 Vectors Suppose a car is heading NE (northeast) at 60 mph. We can use a vector to help draw a picture (see right). v A vector consists of two parts: 1. a magnitude (ex: 60 mph) 2. a direction (ex: NE) By contrast, a scalar only has a magnitude (ex: 60 mph). Examples of vectors include: displacement, velocity, acceleration, and force. Examples of scalars include: distance, speed, time, and volume. A vector has an initial point and a terminal point. The magnitude is the length of the vector. AB Two vectors are equal if they have the same magnitude and direction. So, all of the vectors to the right are equal. To add two vectors visually, put them tip to tail and make a triangle. Or put them tail to tail and make a parallelogram. You can stretch and shrink vectors by multiplying by a scalar. Negative values make the vector go in the opposite direction.

2 Math /7.5 Notes Page 2 of 6 Ex 1. Two forces of 15 and 22 newtons act on a point in the plane. If the angle between the forces is 100, find the magnitude of the resultant force. The equilibrant of a resultant vector is the negative of that vector. So, the equilibrant of the resultant u is u. Ex 2. Two rescue vessels are pulling a broken-down motorboat toward a boathouse with forces of 840 lb and 960 lb. The angle between these forces is Find the direction and magnitude of the equilibrant.

3 Math /7.5 Notes Page 3 of 6 Ex 3. A 3000-lb car is parked on a driveway that is inclined 15 to the horizontal. Find the magnitude of the force required to keep the car from rolling down the driveway. Vectors in a Coordinate System In a two-dimensional coordinate system, we can represent vectors using two components: an x-component (horizontal component) and a y-component (vertical component). Here s the notation: v = a, b The magnitude (or length) of a vector v = a, b is v = a 2 + b 2. The direction angle θ satisfies tan θ = b a. Ex 4. Find the magnitude and direction angle of u = 3, 1. To add/subtract vectors, you just add/subtract their components. ex: 3, 1 + 4,7 = 3 + ( 4), = 1, 6 To multiply a vector by a scalar, you just distribute the scalar to each component. ex: 3 2, 5 = 6, 15 Ex 5. If u = 2, 3 and v = 1,2, find 2u 3v. Notes: 0 = 0,0 is called the zero vector. Vectors with length 1 are called. ex: w = 3 5, 4 5 is a unit vector since w = ( 3 5 )2 + ( 4 5 )2 = 1

4 Math /7.5 Notes Page 4 of 6 Two special unit vectors are given their own letters: i = 1, 0 and j = 0, 1 We can represent any vector in terms of i and j: v = a, b = ai + bj Ex 6. Write u = 5, 8 in terms of i and j. It is often useful to break a vector into its horizontal and vertical components. Ex 7. A vector v has a direction angle of 60 and a magnitude of 8. Find the magnitudes of the horizontal and vertical components, and write v in terms of i and j. Dot Product The dot product of u = a, b and v = c, d is: u v = ac + bd Ex 8. If u = 4, 2 and v = 1,3, find u v. Properties of Dot Products 1. u v = v u 2. (u + v) w = u w + v w 3. (ku ) v = k(u v) = u (kv) 4. u u = u 2 Here are proofs of the first and fourth properties (here, let u = a, b and v = c, d ): 1. u v = ac + bd = ca + db = v u 4. u 2 = ( a 2 + b 2 ) 2 = a 2 + b 2 = u u

5 Math /7.5 Notes Page 5 of 6 The Dot Product Theorem u v = u v cos θ (here, θ is the angle between u and v) The Dot Product Theorem can help us find the angle between two vectors. Ex 9. Find the angle between u = 4, 2 and v = 1,3. Two nonzero vectors u and v are if and only if. Why? If u and v are perpendicular, then θ = π 2 so u v = u v cos π 2 = 0. And if u v = 0, then u v cos θ = 0, so θ must be π and so u and v are perpendicular. 2 Another word for perpendicular is. Ex 10. Determine whether u = 3,5 and v = 2, 8 are orthogonal. Ex 11. Determine whether u = 2,1 and v = 1,2 are orthogonal. Practice 1. A 200-lb box is on a ramp. If a force of 80 lbs is just sufficient to keep the box from sliding, find the angle of inclination of the plane. (Assume no friction on the ramp.)

6 Math /7.5 Notes Page 6 of 6 2. Suppose u = 2, 1 and v = 0,3. a) Find u. c) Write v in terms of i and j. b) Find 5v 2u. 3. Find the magnitudes of the horizontal and vertical components of v given that v = 50 and θ = Find the direction (in degrees) of v = 2, Let u = 3, 1 and v = 2,4. a) Find u v. b) Find the angle between u and v. 6. Determine if u = 2, 5 and v = 10, 4 are orthogonal. Q: A man while looking at a photograph said, "Brothers and sisters have I none. That man's father is my father's son." Who was the person in the photograph?