Unit 6: An Introduction to Vectors
6.1- An Introduction to Vectors
A scalar only has magnitude. Temperature, speed, mass, and volume are examples of scalars.
A vector has magnitude and direction. The magnitude of is written as =v. Position, displacement, velocity and acceleration and force are examples of vector quantities.
Equal Vectors are parallel and have the same magnitude and direction.
Coincident Vectors are equal vectors which can be translated to lie on top of each other.
Opposite Vectors are parallel and have the same magnitude but opposite directions.
Connecting vectors to 2D figures: A vector is always drawn head-to-tail.
6.1 - Questions:
a) State 2 pairs of opposite vectors.
b)State 2 pairs of equivalent vectors.
1. Square ABCD is dra wn as shown below with the diagonals intersecting at E.
2. For the following vector, draw its equal vector and opposite vector.
3. Plot vector AB whose tail is at (4,2) and whose head is at (-2, -4).
4. CD is a vector whose tail is at (1,2) and whose head is at (4,6). Calculate the magnitude of CD.
6.4- Properties if Vectors
Properties of Vector Addition:
1. Commutative Property: a + b = b + a
2. Associative Property: (a + b) + c = a + (b + c)
3. Distributive Property: k(a + b) = k a + k b, k € R
Laws of Vector Addition and Scalar Multiplication:
1. Adding 0 : a + 0 = a
2. Associative Law for Scalars: m(n a ) = (mn)a = m n a
3. Distributive Law for Scalars: (m + n)a = m a + n a
6.4- Questions:
1. Prove the associative law using a diagram. c b
2. Draw the vector a + b + c. a 3. If 4 x + 6 y = a and -2 x + 10 y = 12 b. Express x and y in terms of a and b. 6.7- Operation with Vectors in R³ Representation of vectors in R³: * In R³, i = (1, 0, 0), j = (0, 1, 0) and k = (0, 0, 1). All are unit vectors along the x-, y- and z- axis. * OP= (a, b, c) = ai, bj, ck, OP = a²+b²+c² 6.7 -Questions 1. Write the vector OP = (5, 2, 3) using the standard unit vectors. 2. The vector AB = (1, 2, 3). Determine AB 3. Write the vector SP = 2i, -j, 2k in component form and calculate its magnitude.
