Vectors in the Plane

Vectors in the Plane MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011

Vectors vs. Scalars scalar quantity having only a magnitude (e.g. temperature, volume, length, area) and represented by a real number vector mathematical object having a magnitude and a direction (e.g. velocity, acceleration, force) denoted in boldface v.

Directed Line Segment A directed line segment from initial point A to terminal point B is denoted AB. The magnitude, norm, or length of AB is denoted AB.

Equivalent Vectors Definition Two vectors are equivalent if they have the same direction and magnitude. Notation: u = v

Equivalent Vectors Definition Two vectors are equivalent if they have the same direction and magnitude. Notation: u = v An equivalent vector to v can be produced by rigidly translating the vector without rotation.

Vector Addition A AB + BC = AC A

Scalar Multiplication If we multiply a vector u by a scalar c > 0 we obtain a vector denoted cu which has the same direction as u but length cu = c u.

Scalar Multiplication If we multiply a vector u by a scalar c > 0 we obtain a vector denoted cu which has the same direction as u but length cu = c u. If we multiply a vector u by a scalar c < 0 we obtain a vector denoted cu which has the opposite direction from u but length cu = c u.

Scalar Multiplication If we multiply a vector u by a scalar c > 0 we obtain a vector denoted cu which has the same direction as u but length cu = c u. If we multiply a vector u by a scalar c < 0 we obtain a vector denoted cu which has the opposite direction from u but length cu = c u. Scalar multiplication by c > 1 elongates a vector. Scalar multiplication by c < 1 contracts a vector.

Vector Components Definition Any vector equivalent to the vector with initial point at the origin and terminal point at (a 1, a 2 ) will be denoted a = a 1, a 2. The real numbers a 1 and a 2 are called the components of vector a. a = a 1, a 2 = a1 2 + a2 2 c a = c a 1, a 2 = ca 1, ca 2 a + b = a 1, a 2 + b 1, b 2 = a 1 + b 1, a 2 + b 2

Examples Let a = 1, 2, b = 2, 3, and c = 1/2 and calculate the following. 1 a + b 2 a + ( 1)b 3 a + c b 4 ( c)a + b

Examples Let a = 1, 2, b = 2, 3, and c = 1/2 and calculate the following. 1 a + b = 1, 2 + 2, 3 = 1 + ( 2), 2 + 3 = 1, 5 2 a + ( 1)b = 1, 2 + ( 1) 2, 3 = 1, 2 + 2, 3 = 1 + 2, 2 3 = 3, 1 3 a + c b = 1, 2 + 1 2 2, 3 = 1, 2 + 1, 3 2 = 1 + ( 1), 2 + 3 2 = 0, 7 2 4 ( c)a + b = 1 2 1, 2 + 2, 3 = 1, 1 + 2, 3 = 2 1 2 + ( 2), 1 + 3 = 5 2, 2

Vector Addition Definition The vector 0 = 0, 0 is called the zero vector.

Vector Addition Definition The vector 0 = 0, 0 is called the zero vector. Definition For any vector a, the additive inverse of a is denoted a = ( 1)a = a 1, a 2.

Vector Addition Definition The vector 0 = 0, 0 is called the zero vector. Definition For any vector a, the additive inverse of a is denoted a = ( 1)a = a 1, a 2. Definition Two vectors having the same or opposite direction are called parallel.

Vector Addition Definition The vector 0 = 0, 0 is called the zero vector. Definition For any vector a, the additive inverse of a is denoted a = ( 1)a = a 1, a 2. Definition Two vectors having the same or opposite direction are called parallel. Remark: two vectors are parallel if and only if they are nonzero scalar multiples of each other.

Vector Algebra Theorem For any vectors a, b, c and any scalars d and e, the following hold: 1 a + b = b + a (commutativity) 2 a + (b + c) = (a + b) + c (associativity) 3 a + 0 = a (additive identity) 4 a + ( a) = 0 (additive inverse) 5 d(a + b) = da + db (distributive law) 6 (d + e)a = da + ea (distributive law) 7 1a = a (multiplicative identity) 8 0a = 0 (multiplication by 0)

Standard Basis Vectors We define two vectors: i = 1, 0 and j = 0, 1.

Standard Basis Vectors We define two vectors: i = 1, 0 and j = 0, 1. Note: i = j = 1

Standard Basis Vectors We define two vectors: i = 1, 0 and j = 0, 1. Note: i = j = 1 Any vector a = a 1, a 2 in the plane can be written as a = a 1, a 2 = a 1 i + a 2 j.

Unit Vector Definition Any vector u for which u = 1 is called a unit vector.

Unit Vector Definition Any vector u for which u = 1 is called a unit vector. Theorem For any nonzero vector a = a 1, a 2, a unit vector having the same direction as a is given by u = 1 a a.

Finding a Unit Vector (1 of 2) Example Suppose a = 5, 12. 1 Find a unit vector in the same direction as a. 2 Find a vector twice the length of a and in the same direction as a.

Finding a Unit Vector (1 of 2) Example Suppose a = 5, 12. 1 Find a unit vector in the same direction as a. u = a a = 5, 12 5, 12 = = 5 ( 5) 2 + (12) 2 13 13, 12 13 2 Find a vector twice the length of a and in the same direction as a.

Finding a Unit Vector (1 of 2) Example Suppose a = 5, 12. 1 Find a unit vector in the same direction as a. u = a a = 5, 12 5, 12 = = 5 ( 5) 2 + (12) 2 13 13, 12 13 2 Find a vector twice the length of a and in the same direction as a. w = 2a = 10, 24

Finding a Unit Vector (2 of 2) Example Suppose a = 5, 12. 1 Find a vector of length 2 in the opposite direction of a.

Finding a Unit Vector (2 of 2) Example Suppose a = 5, 12. 1 Find a vector of length 2 in the opposite direction of a. Since u is a unit vector in the same direction as a, then 10 v = 2u = 13, 24 13 is in the opposite direction of a and has length 2.

Polar Form Since every nonzero vector a can be written as a = a a a = a a a = a u where u is a unit vector, then we can develop a polar form for nonzero vectors.

Polar Form Since every nonzero vector a can be written as a = a a a = a a a = a u where u is a unit vector, then we can develop a polar form for nonzero vectors. If u 1 > 0 then let θ = tan 1 u 2 u 1 and a = a cos θ, sin θ.

Polar Form Since every nonzero vector a can be written as a = a a a = a a a = a u where u is a unit vector, then we can develop a polar form for nonzero vectors. If u 1 > 0 then let θ = tan 1 u 2 u 1 If u 1 < 0 then let θ = tan 1 u 2 u 1 and a = a cos θ, sin θ. and a = a cos θ, sin θ.

Polar Form Since every nonzero vector a can be written as a = a a a = a a a = a u where u is a unit vector, then we can develop a polar form for nonzero vectors. If u 1 > 0 then let θ = tan 1 u 2 u 1 If u 1 < 0 then let θ = tan 1 u 2 u 1 and a = a cos θ, sin θ. and a = a cos θ, sin θ. If u 1 = 0 then u = ±j, θ = ±π/2 and a = a 0, ±1.

Example (1 of 2) Suppose a Canada goose is flying northwest at 30 mph in a wind blowing from the south at 20 mph. What is the goose s true course and ground speed?

Example (2 of 2) 40 w 30 g w 20 g 10 Θ g + w = 30 cos 3π 4, sin 3π 4 + 0, 20 = 15 2, 15 2 + 20 g + w 46.3522 mph θ 2.04615 117.236

Homework Read Section 10.1. Exercises: 1 51 odd.